05 ਨਿਰੰਤਰਤਾ ਅਤੇ ਵਿਭਿੰਨਤਾ
ਅਭਿਆਸ 03
Question:
Find dy/dx of ax+by^2=cosy
Answer:
Answer: Step 1: Rewrite the equation in terms of y: ax + by2 = cosy y = (ax + cosy)/b2
Step 2: Differentiate both sides with respect to x: dy/dx = (d(ax + cosy)/dx)/b2
Step 3: Apply the Chain Rule to the left side: dy/dx = (a + (dcosy/dx)*(-1))/b2
Step 4: Differentiate the right side with respect to x: dcosy/dx = -siny
Step 5: Substitute the result from Step 4 into Step 3: dy/dx = (a - siny)/b2
Question:
Find dy/dx of y=sin^−1(1−x^2/1+x^2),0<x<1
Answer:
Answer:
Step 1: Differentiate both sides with respect to x.
dy/dx = d/dx (sin^−1(1−x^2/1+x^2))
Step 2: Use the Chain Rule.
dy/dx = (d/dx (1−x^2/1+x^2)) * (d/dx (sin^−1(1−x^2/1+x^2)))
Step 3: Differentiate the numerator and denominator of the fraction.
dy/dx = [(−2x(1+x^2) - (1−x^2)(2x))/(1+x^2)^2] * (d/dx (sin^−1(1−x^2/1+x^2)))
Step 4: Use the formula for the derivative of inverse sine.
dy/dx = [(−2x(1+x^2) - (1−x^2)(2x))/(1+x^2)^2] * (1/(1-(1−x^2/1+x^2)^2)^(1/2))
Step 5: Simplify the expression.
dy/dx = [(-2x(1+x^2) - (1-x^2)(2x))/(1+x^2)^2] * [1/(1-(1-x^2/1+x^2)^2)^(1/2)]
Step 6: Final Answer.
dy/dx = -2x/(1+x^2)^(3/2)
Question:
Find dy/dx of sin^2x+cos^2y=1
Answer:
Answer:
Step 1: Take the derivative of each side of the equation with respect to x.
d/dx (sin2x + cos2y) = d/dx (1)
Step 2: Apply the chain rule on the left side of the equation.
2*sin(2x)cos(2x) + 0cos(2y) = 0
Step 3: Simplify the equation.
2*sin(2x)*cos(2x) = 0
Step 4: Factor out the 2*sin(2x).
2sin(2x)[cos(2x)] = 0
Step 5: Solve for the derivative.
2*sin(2x) = 0
sin(2x) = 0
2x = (2n + 1)π/2 (where n is an integer)
x = (n + 1/2)π (where n is an integer)
Question:
Find dy/dx of x^3+x^2y+xy^2+y^3=81
Answer:
Answer:
Step 1: Rewrite the equation as a function of y: y^3 + y^2(x+1) + y(x^2+1) + x^3 = 81
Step 2: Take the derivative of both sides with respect to x: 3x^2 + 2xy + y^2(1) + y(2x) = 0
Step 3: Simplify the equation: 3x^2 + 2xy + y^2 + 2xy = 0
Step 4: Factor out y from the equation: y(3x + 2x + 1) + y(2x + 1) = 0
Step 5: Divide both sides of the equation by (3x + 2x + 1): y + 2xy/(3x + 2x + 1) = 0
Step 6: Take the derivative of both sides with respect to x: dy/dx = -2y/(3x + 2x + 1)^2
Question:
Find dy/dx of y=tan^−1(3x−x^3/1−3x^2),−1/√3<x<1/√3
Answer:
Step 1: Rewrite y in terms of inverse tangent: y = tan^−1(3x−x^3/1−3x^2)
Step 2: Take the derivative of both sides with respect to x: dy/dx = (3 - 3x^2)/(1 - 3x^2)^2 * d/dx(3x - x^3)
Step 3: Take the derivative of the numerator: dy/dx = (3 - 3x^2)/(1 - 3x^2)^2 * (3 - 3x^2)
Step 4: Simplify: dy/dx = (3 - 3x^2)^2/(1 - 3x^2)^2
Question:
Find dy/dx of y=sin^−1(2x/1+x^2)
Answer:
Answer: Step 1: Rewrite the equation as y = arcsin(2x/(1+x^2)) Step 2: Take the derivative of both sides of the equation with respect to x, using the chain rule: dy/dx = (2/(1+x^2))(1) - (2x2x)/(1+x^2)^2 Step 3: Simplify the equation: dy/dx = 2/(1+x^2)^2 - 4x/(1+x^2)^2 Step 4: Simplify further: dy/dx = (2 - 4x)/(1+x^2)^2
Question:
Find dy/dx of 2x+3y=sinx
Answer:
Step 1: Rewrite the equation in terms of y: 2x + 3y = sin x 3y = sin x - 2x y = (sin x - 2x)/3
Step 2: Take the derivative of both sides with respect to x: dy/dx = -2/3 - (cos x)/3
Therefore, dy/dx = -2/3 - (cos x)/3
Question:
Find dy/dx of y=cos^−1(2x/1+x^2),−1<x<1
Answer:
Answer: Step 1: Differentiate y with respect to x using chain rule dy/dx = - (2/(1 + x^2)^2) * (1 + 2x^2)
Step 2: Simplify the expression dy/dx = - (2/(1 + x^2)^2) * (1 + 2x^2) = - (2/(1 + x^2)^2) * (1 + 2x^2) = - (2/(1 + x^2)^2) * (1 + 2x^2) = - (2/(1 + x^2)) * (2x) = - (4x)/(1 + x^2)
Question:
Find dy/dx of sin^2y+cosxy=k
Answer:
Answer: Step 1: Take the derivative of both sides of the equation with respect to x.
dy/dx = (2sin(y)cos(y) + cos(x)y)dx - 0
Step 2: Simplify the equation
dy/dx = (2sin(y)cos(y) + cos(x)y)dx
Step 3: Factor out dy
dy/dx = dy(2sin(y)cos(y) + cos(x)y)dx
Step 4: Divide both sides by (2sin(y)cos(y) + cos(x)y)
dy/dx = dy/ (2sin(y)cos(y) + cos(x)y)dx
Therefore, the answer is dy/dx = dy/ (2sin(y)cos(y) + cos(x)y)dx
Question:
Find dy/dx of y=sin^−1(2x√1−x^2),−1/√2<x<1/√2
Answer:
Step 1: Rewrite the equation in terms of x and y: y = sin^−1(2x√1−x^2)
Step 2: Take the derivative with respect to x: dy/dx = (1/√1−x^2) * (2√1−x^2 * (1/2) - 2x * (2x))
Step 3: Simplify: dy/dx = (1/√1−x^2) * (1 - 4x^2)
Question:
Find dy/dx of x^2+xy+y^2=100
Answer:
Step 1: Rewrite the equation as y = f(x) = (100-x^2)/x
Step 2: Differentiate the equation with respect to x using the Chain Rule:
dy/dx = (2x(100-x^2) - (100-x^2))/x^2
Step 3: Simplify the equation:
dy/dx = (200x - x^3 - 100)/x^2
Question:
Find dy/dx of 2x+3y=siny
Answer:
Answer:
-
Rewrite the equation in the form y = f(x): 3y = siny - 2x y = (siny - 2x) / 3
-
Differentiate both sides with respect to x: dy/dx = (cosy - 2) / 3
Question:
Find dy/dx of xy+y^2=tanx+y
Answer:
Step 1: Rewrite the equation in the form of y in terms of x.
y = tanx - xy + y^2
Step 2: Take the derivative of both sides with respect to x.
dy/dx = (tanx)’ - (xy)’ + (y^2)'
Step 3: Simplify the equation.
dy/dx = sec^2x - y - 2yy'
Step 4: Substitute y = tanx - xy + y^2 into the equation.
dy/dx = sec^2x - tanx + xy - 2y(tanx - xy + y^2)'
Step 5: Take the derivative of both sides with respect to x.
dy/dx = sec^2x - tanx + xy - 2y(1 - x - 2yy')
Step 6: Simplify the equation.
dy/dx = sec^2x - tanx + xy - 2y + 2xyy'
Question:
Find dy/dx of y=sec^−1(1/2x^2−1),0<x<1/√2
Answer:
Answer: Step 1: Rewrite the equation in terms of y y = arcsec(1/2x^2 - 1)
Step 2: Take the derivative of both sides with respect to x dy/dx = (1/2x^2 - 1)’ * (2x)
Step 3: Simplify the expression dy/dx = x / (2x^2 - 2)
Question:
Find dy/dx of y=cos^−1(1−x^2/1+x^2),0<x<1
Answer:
Answer: Step 1: Let y = cos^-1(1 - x^2/1 + x^2)
Step 2: Differentiate both sides with respect to x
Step 3: dy/dx = -2x/(1 + x^2)^2 * (-1)
Step 4: dy/dx = 2x/(1 + x^2)^2
Hence, dy/dx = 2x/(1 + x^2)^2, 0 < x < 1
ਜੇਈਈ ਅਧਿਐਨ ਸਮੱਗਰੀ (ਗਣਿਤ)
01 ਸਬੰਧ ਅਤੇ ਕਾਰਜ
02 ਉਲਟ ਤਿਕੋਣਮਿਤੀ ਫੰਕਸ਼ਨ
03 ਮੈਟ੍ਰਿਕਸ
04 ਨਿਰਧਾਰਕ
05 ਨਿਰੰਤਰਤਾ ਅਤੇ ਵਿਭਿੰਨਤਾ
06 ਡੈਰੀਵੇਟਿਵਜ਼ ਦੀ ਐਪਲੀਕੇਸ਼ਨ
07 ਅਟੁੱਟ
08 ਇੰਟੀਗ੍ਰੇਲਸ ਦੀ ਐਪਲੀਕੇਸ਼ਨ
09 ਵੈਕਟਰ
10 ਤਿੰਨ ਅਯਾਮੀ ਜਿਓਮੈਟਰੀ
11 ਲੀਨੀਅਰ ਪ੍ਰੋਗਰਾਮਿੰਗ
12 ਸੰਭਾਵਨਾ