Question:

Find derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (x^2+1)cos x

Answer:

1. Use the chain rule to calculate the derivative:

d/dx[(x^2 + 1)cos x] = d/dx[(x^2 + 1)]cos x + (x^2 + 1)d/dx[cos x]

2. Calculate the derivative of the first term:

d/dx[(x^2 + 1)] = 2x

3. Calculate the derivative of the second term:

d/dx[cos x] = -sin x

4. Substitute the derivatives into the equation:

d/dx[(x^2 + 1)cos x] = 2xcos x - (x^2 + 1)sin x

Question:

Find the derivative of the following functions from first principle: cos(x−π/8)

Answer:

Step 1: Identify the function, f(x) = cos(x−π/8).

Step 2: Use the definition of the derivative to calculate the derivative of f(x):

df/dx = lim h→0 (cos(x-π/8 + h) - cos(x-π/8))/h

Step 3: Use the sum and difference identities for cosine to simplify the equation:

df/dx = lim h→0 (cos(x+h-π/8) - cos(x-π/8))/h

Step 4: Use the product to sum identity for cosine to further simplify the equation:

df/dx = lim h→0 (cos(x-π/8)cos(h) - cos(x-π/8)cos(h))/h

Step 5: Simplify the equation:

df/dx = lim h→0 (cos(x-π/8)(1 - cos(h))/h

Step 6: Use the power reduction identity for cosine to simplify the equation:

df/dx = lim h→0 (cos(x-π/8)(2sin2(h/2))/h

Step 7: Use the double angle identity for sin to simplify the equation:

df/dx = lim h→0 (cos(x-π/8)(2sin(h)cos(h))/h

Step 8: Use the limit definition of the derivative to calculate the derivative of f(x):

df/dx = lim h→0 (2sin(h)cos(x-π/8))/h = 2cos(x-π/8)lim h→0 (sin(h))/h = 2cos(x-π/8)·1 = 2cos(x-π/8)

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers : (x+a)

Answer:

Answer:

1. First, use the power rule: Derivative of (x+a) = d/dx (x+a) = 1

2. Since a is a constant, the derivative is equal to 1. Therefore, the derivative of (x+a) = 1

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,r and s are fixed non-zero constants and m and n are integers) : a/x^4−b/x^2+cos x

Answer:

Answer: Derivative of a/x^4 = -4a/x^5 Derivative of b/x^2 = -2b/x^3 Derivative of cos x = -sin x

Therefore, the derivative of the given function is -4a/x^5 -2b/x^3 -sin x.

Question:

Find the derivative of the following functions(it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : cosx/(1+sin x)

Answer:

Step 1: Rewrite the given function as cos x/1+sin x

Step 2: Use the quotient rule to find the derivative of the function

Derivative = (1+sin x)(-cos x) - (cos x)(-sin x) / (1+sin x)^2

Step 3: Simplify the derivative

Derivative = -cos^2 x - sin^2 x / (1+sin x)^2

Step 4: Use the double angle identity to simplify the derivative

Derivative = -1 / (1+sin x)^2

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : cosec x cot x

Answer:

Step 1: Use the product rule to find the derivative of cosec x cot x.

Step 2: Differentiate cosec x with respect to x.

Step 3: Differentiate cot x with respect to x.

Step 4: Multiply the derivatives of cosec x and cot x.

Step 5: Simplify the expression.

Answer: The derivative of cosec x cot x is -cosec x cosec x.

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : sin^n x

Answer:

1. Use the chain rule: d/dx (sin^n x) = n sin^(n-1) x cos x

2. Simplify: n sin^(n-1) x cos x

Question:

Find the derivative of the following functions(it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (ax+b)/(cx+d)

Answer:

Step 1: Rewrite the function in the form of a fraction (ax+b)/(cx+d)

Step 2: Apply the quotient rule

Derivative = [(c(ax+b)) - (d(cx+d))]/[(cx+d)^2]

Step 3: Simplify the expression

Derivative = [(acx+bc) - (dcx+bd)]/[(cx+d)^2]

Step 4: Simplify further

Derivative = (ad-bc)/[(cx+d)^2]

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (sec x−1)/(sec x+1)

Answer:

Answer:

1. First, use the quotient rule to find the derivative of (sec x - 1)/(sec x + 1):

d/dx [(sec x - 1)/(sec x + 1)] = (sec x + 1)(d/dx sec x) - (sec x - 1)(d/dx sec x) / (sec x + 1)^2

2. Now, use the chain rule to find the derivative of sec x:

d/dx sec x = sec x tan x (d/dx x)

3. Substitute this into the equation from step 1:

d/dx [(sec x - 1)/(sec x + 1)] = (sec x + 1)(sec x tan x)(d/dx x) - (sec x - 1)(sec x tan x)(d/dx x) / (sec x + 1)^2

4. Simplify the equation:

d/dx [(sec x - 1)/(sec x + 1)] = (sec x tan x)(d/dx x) / (sec x + 1)^2

5. Finally, use the power rule to simplify the equation further:

d/dx [(sec x - 1)/(sec x + 1)] = (sec^2 x tan x)(d/dx x) / (sec^2 x + 1)

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (ax+b)(cx+d)^2

Answer:

Answer:

Step 1: Rewrite the function as: (ax+b)(cx^2 + 2dx + d^2)

Step 2: Apply the product rule: d/dx [(ax+b)(cx^2 + 2dx + d^2)] = (ax+b) * (2cx + 2d) + (cx^2 + 2dx + d^2) * (a)

Step 3: Simplify the expression: d/dx [(ax+b)(cx^2 + 2dx + d^2)] = 2acx^2 + (2ad + 2bc)x + (2bd + ab)

Question:

Find the derivative of the following function from first principle: −x

Answer:

Step 1: Identify the function

Function: f(x) = -x

Step 2: Compute the difference quotient

Difference Quotient: (f(x + h) - f(x))/h

Step 3: Simplify the difference quotient

Difference Quotient: (-(x + h) - (-x))/h

Step 4: Take the limit of the difference quotient as h approaches 0

Limit of Difference Quotient: lim h→0 (-(x + h) - (-x))/h

Step 5: Simplify the limit

Limit of Difference Quotient: lim h→0 (-h)/h

Step 6: Compute the limit

Limit of Difference Quotient: lim h→0 -1

Step 7: The derivative of the function is

Derivative of f(x) = -x: -1

Question:

Find the derivative of the following functions form first principle: (−x)^(−1)

Answer:

Answer: Step 1: Let f(x) = (−x)^(−1)

Step 2: Find f’(x) using the first principle: f’(x) = lim h->0 [f(x+h) - f(x)]/h

Step 3: Substitute the function f(x) in the equation: f’(x) = lim h->0 [((−x+h)^(−1)) - (−x)^(−1)]/h

Step 4: Simplify the equation: f’(x) = lim h->0 [((−1/x+h)^(−1)) - (−1/x)^(−1)]/h

Step 5: Simplify further: f’(x) = lim h->0 [1/(−1/x+h) - 1/(−1/x)]/h

Step 6: Solve for f’(x): f’(x) = lim h->0 [(-x - hx)/(x + hx)(-x)]/h

Step 7: Simplify: f’(x) = lim h->0 [(-1)/(x + hx)]/h

Step 8: Solve for f’(x): f’(x) = lim h->0 [-1/(x + hx)]/h

Step 9: Simplify further: f’(x) = -1/x^2

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (a+b sin x)/(c+d cos x)

Answer:

1. Rewrite the function as: (a+b sin x)(c-d cos x) / (c+d cos x)(c+d cos x)

2. Take the derivative of each term in the numerator and denominator: Numerator: (a+b sin x)’ (c-d cos x) + (c-d cos x)’ (a+b sin x) Denominator: (c+d cos x)’ (c+d cos x)

3. Simplify the derivatives: Numerator: b cos x (c-d cos x) - d sin x (a+b sin x) Denominator: 2d cos x (c+d cos x)

4. Simplify the fraction: (b cos x (c-d cos x) - d sin x (a+b sin x)) / (2d cos x (c+d cos x))

Question:

Find the derivative of the following functions(it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (px^2+qx+r)/(ax+b)

Answer:

Step 1: Rewrite the function as: (px^2 + qx + r)(ax + b)^-1

Step 2: Take the derivative of both sides: 2px(ax + b)^-1 + q(ax + b)^-1 - (px^2 + qx + r)(ax + b)^-2(-a)

Step 3: Simplify the expression: 2px(ax + b)^-1 + q(ax + b)^-1 + (px^2 + qx + r)(-a)(ax + b)^-2

Step 4: Simplify further: 2px(ax + b)^-1 + q(ax + b)^-1 - (pa^2x^2 + qax + ra)(ax + b)^-2

Question:

Find the derivative of the following fucctions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (1+1/x)/(1−1/x)

Answer:

Given, f(x) = (1+1/x)/(1−1/x)

Step 1: Rewrite the function as

f(x) = (1+1/x) (1/1−1/x)

Step 2: Take the derivative of both sides

f’(x) = (1/x) (1/1−1/x) + (1+1/x) (-1/x2) (1/1−1/x)

Step 3: Simplify the equation

f’(x) = (1/x2)(1+1/x−1−1/x)

Step 4: Simplify further

f’(x) = (1/x2)(2−1/x)

Step 5: Final answer

f’(x) = (2−1/x)/x2

Question:

Find the derivative of the following functions from first principle: sin(x+1)

Answer:

1. Rewrite the function as f(x) = sin(x+1)

2. Take the derivative of f(x) using the definition of the derivative: f’(x) = lim h->0 (sin(x+1+h) - sin(x+1))/h

3. Simplify the expression: f’(x) = lim h->0 (sin(x+h+1) - sin(x+1))/h

4. Use the identity sin(a+b) = sin(a)cos(b) + cos(a)sin(b): f’(x) = lim h->0 (sin(x+1)cos(h) + cos(x+1)sin(h) - sin(x+1))/h

5. Simplify the expression: f’(x) = lim h->0 (cos(x+1)sin(h))/h

6. Use the identity sin(a) = a when a is small: f’(x) = lim h->0 (cos(x+1)h)/h

7. Simplify the expression: f’(x) = lim h->0 cos(x+1)

8. Evaluate the limit: f’(x) = cos(x+1)

Question:

Find the derivative of the following function (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (ax+b)/(cx+d)

Answer:

Given: f(x) = (ax+b)/(cx+d)

Step 1: Rewrite the function in the form of f(x) = u/v

f(x) = (ax+b)/(cx+d) = (ax+b)/v, where v = (cx+d)

Step 2: Apply the quotient rule to calculate the derivative of the function.

f’(x) = [v(du/dx) - u(dv/dx)]/v^2

Step 3: Substitute the values of u and v in the equation.

f’(x) = [v(a) - (ax+b)(c)]/v^2

Step 4: Simplify the equation.

f’(x) = [cx+d)(a) - (ax+b)(c)]/[(cx+d)^2]

f’(x) = [acx+ad - acx - bc]/[(cx+d)^2]

f’(x) = ad/[(cx+d)^2]

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (ax^2+sin x)(p+q cos x)

Answer:

Step 1: Use the Product Rule to differentiate the given expression.

Step 2: Differentiate the first factor, (ax^2 + sin x), using the Power Rule.

Step 3: Differentiate the second factor, (p + q cos x), using the Chain Rule.

Step 4: Combine the two derivatives to obtain the final answer.

Answer: (2ax + cos x)(p + q cos x) + (ax^2 + sin x)(-q sin x)

Question:

If the derivative of the function 4√x−2 is a/√x. Find the value of a.

Answer:

1. Use the definition of a derivative to find a:

a = (4√x - 2)’ = 4/2√x

2. Substitute x = 1 in the equation to solve for a:

a = 4/2√1

3. Simplify to find the value of a:

a = 4

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) :(4x+5 sin x)/(3x+7 cosx)

Answer:

1. Rewrite the function in its simplest form:

(4x + 5sin x) / (3x + 7cos x)

2. Use the Quotient Rule to find the derivative:

[(3x + 7cos x)(4) - (4x + 5sin x)(-7cos x)] / (3x + 7cos x)^2

3. Simplify the derivative:

[12x + 28cos x + 28x - 35sin x] / (3x + 7cos x)^2

4. Simplify further:

[40x + 28cos x - 35sin x] / (3x + 7cos x)^2

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) :x/(1+tan x)

Answer:

Step 1: Rewrite the function as x/1+tan x

Step 2: Apply the quotient rule to the function

Step 3: Find the derivative of the numerator

Derivative of numerator = 1

Step 4: Find the derivative of the denominator

Derivative of denominator = sec^2 x

Step 5: Substitute the derivatives of the numerator and denominator into the quotient rule

Derivative of x/(1+tan x) = 1/sec^2 x - x sec^2 x/(1+tan x)^2

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r,s and s are fixed non-zero constants and m and n are integers) : x^4(5 sin x−3 cos x)

Answer:

Answer: Step 1: Take the derivative of x^4 Derivative of x^4 = 4x^3

Step 2: Take the derivative of 5 sin x Derivative of 5 sin x = 5 cos x

Step 3: Take the derivative of -3 cos x Derivative of -3 cos x = -3 (-sin x)

Step 4: Combine the derivatives Derivative of x^4 (5 sin x - 3 cos x) = 4x^3 (5 cos x - 3 (-sin x))

Step 5: Simplify Derivative of x^4 (5 sin x - 3 cos x) = 20x^3 sin x + 9x^3 cos x

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : sin(x+a)/cosx

Answer:

Step 1: Rewrite the given expression as sin(x+a) * cosx^-1.

Step 2: Apply the quotient rule for derivatives.

Step 3: Derivative of sin(x+a) = cos(x+a)

Step 4: Derivative of cosx^-1 = -cosx^-2

Step 5: Multiply the derivatives of the numerator and denominator.

Step 6: Simplify the expression.

Answer: The derivative of sin(x+a)/cosx is cos(x+a) * -cosx^-2 = -sin(x+a)cosx^-2.

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (x+cosx)(x−tanx)

Answer:

1. Use the product rule to find the derivative of (x + cosx)(x − tanx):

d/dx [(x + cosx)(x − tanx)] = (x + cosx)d/dx (x − tanx) + (x − tanx)d/dx (x + cosx)

2. Find the derivative of the first term:

(x + cosx)d/dx (x − tanx) = (x + cosx)(1 − sec2x)

3. Find the derivative of the second term:

(x − tanx)d/dx (x + cosx) = (x − tanx)(1 + cosx)

4. Add the two derivatives together:

d/dx [(x + cosx)(x − tanx)] = (x + cosx)(1 − sec2x) + (x − tanx)(1 + cosx)

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constant and m and n are integers) : (ax+b)^n

Answer:

Answer:

Step 1: Use the power rule to find the derivative.

Derivative = n(ax + b)^(n-1) * a

Step 2: Simplify the derivative.

Derivative = an(ax + b)^(n-1)

Question:

Find the derivative of the following function (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (x^2cos(π/4))/sin x

Answer:

Given, f(x) = (x^2cos(π/4))/sin x

Step 1: Rewrite the function using the product rule

f(x) = (x^2cos(π/4))/sin x = x^2cos(π/4) . (1/sin x)

Step 2: Apply the product rule

f’(x) = [2xcos(π/4) . (1/sin x)] + [x^2cos(π/4) . (-cos x/sin^2 x)]

Step 3: Simplify the expression

f’(x) = [2xcos(π/4) . (1/sin x)] + [x^2cos(π/4) . (-cos x/sin^2 x)] = (2xcos(π/4))/sin x - (x^2cos(π/4)cos x)/sin^3 x

Hence, the derivative of the given function is f’(x) = (2xcos(π/4))/sin x - (x^2cos(π/4)cos x)/sin^3 x

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (px+q)(r/x+s)

Answer:

Step 1: Rewrite the function in terms of multiplication and division:

px + q × (r/x + s)

Step 2: Use the product rule to differentiate the function:

(px + q)’ × (r/x + s) + (px + q) × (r/x + s)'

Step 3: Differentiate each term:

p + q’ × (r/x + s) + (px + q) × (r/x-2s)

Step 4: Simplify:

p + q’ × (r/x + s) + (px + q) × (r/x-2s)

= p + qr/x2 - 2qs + pxr/x2 - 2qsr/x2

Step 5: Simplify further:

p + qr/x2 - 2qs + pxr/x2 - 2qsr/x2

= p + qr/x2 + pxr/x2 - 2qs - 2qsr/x2

= p(1 + xr/x2) + q(r/x2 - 2sr/x2)

= p + qr/x2 - 2qs/x2

Question:

Find the derivative of the following functions(it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers): 1/(ax^2+bx+c)

Answer:

1. Rewrite the function in the form of y = (d/dx)(ax^2+bx+c)

2. Differentiate both sides of the equation with respect to x

y’ = (d/dx)(2ax + b)

3. Substitute the values of a, b and c

y’ = (d/dx)(2a*x + b)

4. Simplify the expression

y’ = 2a

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : sin(x+a)

Answer:

1. Use the Chain Rule: d/dx[sin(x+a)] = d/du[sin(u)] * du/dx[x+a]

2. Differentiate the inside function: d/du[sin(u)] = cos(u)

3. Differentiate the outside function: du/dx[x+a] = 1

4. Substitute: d/dx[sin(x+a)] = cos(x+a)

Question:

Find the derivative of the following (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (sin x+cos x)/(sinx−cosx)

Answer:

Answer: Step 1: Rewrite the expression as: (sin x+cos x)/(sinx−cosx) = (a sin x + b cos x)/(c sin x − d cos x) Step 2: Use the Quotient Rule to find the derivative: d/dx [(a sin x + b cos x)/(c sin x − d cos x)] = (p sin x + q cos x)(c sin x − d cos x) - (a sin x + b cos x)(r sin x + s cos x)/[(c sin x − d cos x)^2] Step 3: Simplify the expression: d/dx [(a sin x + b cos x)/(c sin x − d cos x)] = (p sin x + q cos x)(c sin x − d cos x) - (a sin x + b cos x)(r sin x + s cos x)/[(c sin x − d cos x)^2] = (p c - q d) sin x cos x + (q c - p d) sin^2 x - (a r + b s) sin x cos x - (b r + a s) cos^2 x/[(c sin x − d cos x)^2]

Question:

Find the derivative of the following fuctions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : x/sin^2x

Answer:

Step 1: Rewrite the function as x/sin^2x = x(sin^2x)^-1

Step 2: Take the derivative of both sides of the equation using the chain rule: d/dx[x(sin^2x)^-1] = (sin^2x)^-1d/dx[x] + xd/dx[(sin^2x)^-1]

Step 3: Simplify the equation on the right hand side: d/dx[x(sin^2x)^-1] = (sin^2x)^-11 + x(-2sinxcosx)*(sin^2x)^-2

Step 4: Simplify the equation further: d/dx[x(sin^2x)^-1] = (sin^2x)^-1 - 2xsinxcosx*(sin^2x)^-2

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are fixed integers) : (x+secx)(x−tanx)

Answer:

Step 1: Rewrite the function as: (x + secx)*(x - tanx)

Step 2: Use the Product Rule to find the derivative of the function: d/dx [(x + secx)(x - tanx)] = (d/dx [x + secx])(x - tanx) + (x + secx)*(d/dx [x - tanx])

Step 3: Use the Chain Rule to find the derivatives of the individual terms: d/dx [x + secx] = 1 + (d/dx [secx]) d/dx [x - tanx] = 1 - (d/dx [tanx])

Step 4: Use the Chain Rule to find the derivatives of secx and tanx: d/dx [secx] = secx*tanx d/dx [tanx] = sec^2x

Step 5: Substitute the derivatives of secx and tanx into the equation: d/dx [(x + secx)(x - tanx)] = (1 + secxtanx)(x - tanx) + (x + secx)(1 - sec^2x)

Step 6: Simplify the equation: d/dx [(x + secx)(x - tanx)] = x - tanx + secxtanx + x - secx*sec^2x

Step 7: Simplify the equation further: d/dx [(x + secx)(x - tanx)] = 2x - (tanx + secxsec^2x)

Question:

Find the derivative of the following functions (it is to be understood that a,b,c,d,p,q,r and s are fixed non-zero constants and m and n are integers) : (ax+b)^n(cx+d)^m

Answer:

Answer:

1. Use the power rule to expand the function: Derivative of (ax + b)^n = n(ax + b)^(n-1) * (a) Derivative of (cx + d)^m = m(cx + d)^(m-1) * (c)

2. Multiply the two derivatives: n(ax + b)^(n-1) * (a) * m(cx + d)^(m-1) * (c)

3. Simplify the expression: nm(ax + b)^(n-1) * (cx + d)^(m-1) * (ac)