Differentiation
Differentiation is a method to find rates of change and is an important topic for the JEE exam. The derivative of a function $y = f(x)$ of a variable $x$ is the rate of change of $y$ with respect to the rate of change of $x$. This article helps you to learn about the derivative of a function, standard derivatives, theorems of derivatives, differentiation of implicit functions, and higher order derivatives, all accompanied by solved examples.
Related Topics:
Limits, Continuity and Differentiability
How to Differentiate a Function
The differentiation of a function is a way to calculate the rate of change of a function at a given point. For real-valued functions, it is the slope of the tangent line at a point on a graph.
The derivative of $y$ with respect to $x$ is often written as $\frac{\text{d}y}{\text{d}x}$ and is defined as the limit of the change in $y$ over the change in $x$ as the distance between $x_0$ and $x_1$ approaches zero, or infinitesimally.
In mathematical terms,
The derivative of $f$ at $a$, denoted $f’(a)$, is defined as: f′(a)=limh→0f(a+h)−f(a)h
(f′(a)=limh→0f(a+h)−f(a)h)
Standard Derivatives
(1] ddx(un)=nun−1dudx )
(\frac{d}{dx}\left(c\right)=0,; \text{where c is a constant})
(dydx=∂y∂u∂u∂x where y=F(u) and u=f(x))
Chain Rule (or Function of a Function Rule)
4. Derivatives of Trigonometric Functions
(ddx(sinu)=cosududx ddx(cosu)=−sinududx ddx(tanu)=sec2ududx ddx(secu)=secutanududx ddx(cosec u)=−cosecu cotu dudx ddx(cot u)=−cosec2u dudx )
5. Derivatives of Inverse Trigonometric Functions
(ddx(sin−1u)=1√1−u2dudx,−1<u<1 ddx(cos−1u)=−1√1−u2dudx,−1<u<1 ddx(tan−1u)=11+u2dudx ddx cosec−1u=−1|u|√u2−1dudx|u|>1 ddx(sec−1u)=1|u|√u2−1dudx|u|>1 ddx(cot−1u)=−11+u2dudx )
6. Exponential and Logarithmic Functions
(ddx(eu)=eududx ddx(au)=aulnadudx,where a>0,a≠1 ddx(lnu)=1ududx ddx(lnau)=1ulnadudx, where a>0,a≠1 )
Hyperbolic Functions
**(\begin{array}{l} \frac{d}{dx}\left(\sinh u\right)=\cosh u\frac{du}{dx}\ \frac{d}{dx}\left(\cosh u\right)=\sinh u\frac{du}{dx}\ \frac{d}{dx}\left(\tanh u\right)=sech^{2}u\frac{du}{dx}\ \frac{d}{dx}\ (sech u)=-sechu\ tanhu\frac{du}{dx}\ \frac{d}{dx}\ (cosech u)=-cosechu\ cothu\frac{du}{dx}\ \frac{d}{dx}\ (coth u)=-cosech^{2}u\frac{du}{dx}\\end{array})
Inverse Hyperbolic Functions :eight:
$$\begin{array}{l}
\frac{d}{dx}\sinh^{-1}u = \frac{1}{\sqrt{1 + u^2}}\frac{du}{dx} \\
\frac{d}{dx}\cosh^{-1}u = \frac{1}{\sqrt{u^2 - 1}}\frac{du}{dx}, \quad u > 1 \\
\frac{d}{dx}\tanh^{-1}u = \frac{1}{1 - u^2}\frac{du}{dx}, \quad |u| < 1 \\
\frac{d}{dx}\cosech^{-1}u = -\frac{1}{|u|\sqrt{u^2 + 1}}\frac{du}{dx}, \quad u \ne 0 \\
\frac{d}{dx}\sech^{-1}u = -\frac{1}{u\sqrt{1 - u^2}}\frac{du}{dx}, \quad 0 < u < 1 \\
\frac{d}{dx}\coth^{-1}u = \frac{1}{1 - u^2}\frac{du}{dx}, \quad |u| > 1
\end{array}$$
Some Standard Substitution
Expression Substitution
Simple tricks to solve complicated differential equations are listed below:
If function contains $\sqrt{a^2 - x^2}$; then substitute $x = a \sin\theta$ or $x = a \cos\theta$
(ii) If function contains $\sqrt{{{a}^{2}}+{{x}^{2}}}$; then substitute $x = a \cot{\theta}$ or $x = a \tan{\theta}$
(iii) If function contains $\sqrt{{{a}^{2}}+{{a}^{2}}\sec^2\theta}$ or $\sqrt{{{a}^{2}}+{{a}^{2}}\csc^2\theta}$ ; then substitute $x = a \cosec\theta$ or $x = a \sec \theta$
Theorems of Derivatives
Find below some of the important theorem results:
(ddx[u±v]=dudx±dvdx ddxuv=udvdx+vdudx(Product Rule) ddxuv=vdudx−udvdxv2(Quotient Rule) ddx(k,f(x))=kddx(f(x)),,,where,k,is constant)
Answer:
dydx=xcosxlogx+sinx(1+logxx)
Solution: y′=1×sinxlogx+xcosxlogx+xsinx×1x=sinxlogx+xcosxlogx+sinx
Example 2: Find $\frac{dy}{dx}$ for $y = \sin(x^2 + 1)$.
Solution: y′=cos(x2+1)×2x
2*cos(x^2 + 1)
Differentiation of Implicit Functions
In Implicit Differentiation, also known as the Chain Rule, differentiate both sides of an equation with two given variables by considering one of the variables as a function of the second variable. In other words, differentiate the given function with respect to x and solve for dy/dx. Let us take a look at some examples.
Answer:
Example 1: Find $\frac{dy}{dx}$ if $x^2 + 2xy + y^3 = 4$.
Solution: Differentiating both sides w.r.t. x, we get ddx(x2)+2ddx(xy)+ddx(y3)=ddx(4)
(2xdydx+2xdydx+2y+3y2dydx=0)
(\frac{d}{dx}\left(y\right)=-\frac{2\left(x+y\right)}{2x+3y^2})
Answer:
dd√cosxlogsinx=cosxsinx√cosx
Solution: Let u = log sin x and v = (\sqrt{\cos x})
(dudx=cotx dvdx=−sinx2√cosx)
(\frac{du}{dx}=-2\sqrt{\cos x}\cot x\csc x)
Higher Order Derivatives
Differentiation process can be continued up to the nth derivative of a function. Generally, we deal with the first-order and second-order derivatives of the functions.
\frac{dy}{dx}
is the first derivative of $y$ with respect to $x$
The second derivative of y with respect to x is d2y/dx2.
Similarly, finding the third, fourth, fifth and higher-order derivatives of any function, say g(x), are known as successive derivatives of g(x).
gn(x)
or d^n/dx^n
Proof: Let (u=\tan^{-1} x)
Then (y=e^u)
(\frac{d}{dx} u = \frac{1}{1+x^2})
(\frac{dy}{dx} = \frac{d}{dx} e^u = e^u \frac{d}{dx} u = e^u \frac{1}{1+x^2})
(\frac{d^2 y}{dx^2} = \frac{d}{dx} \left(e^u \frac{1}{1+x^2}\right) = e^u \frac{d}{dx} \frac{1}{1+x^2} + \frac{1}{1+x^2} \frac{d}{dx} e^u)
(\frac{d^2 y}{dx^2} = e^u \frac{-2x}{(1+x^2)^2} + \frac{1}{1+x^2} e^u \frac{1}{1+x^2})
(\frac{d^2 y}{dx^2} = e^u \frac{1-2x}{(1+x^2)^2})
(\frac{d^2 y}{dx^2} = (1-2x) \frac{dy}{dx})
Given:
This is a heading
Solution:
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(etan−1x=y)
$$\frac{dy}{dx} = e^{\tan^{-1}x}\frac{1}{1+x^2}$$
(\frac{e^{\tan^{-1}x}}{1+x^{2}}) (i)
(\frac{d^2y}{dx^2} = \frac{(1+x^2)e^{\tan^{-1}x}\frac{1}{1+x^2} - 2xe^{\tan^{-1}x}}{(1+x^2)^2})
$$\frac{(1-2x)e^{\tan^{-1}x}}{(1+x^{2})^{2}}$$
(\frac{d^2y}{dx^2} \left(1 + x^2\right) = \frac{(1-2x)e^{\tan^{-1}x}}{\left(1+x^{2}\right)})
\frac{d}{dx}\left( (1-2x)y \right)
(from eqn (i))
Therefore, it is proven.
Video Lessons
Methods of Differentiation – JEE Solved Questions
Important Theorems of Differentiation for JEE
Frequently Asked Questions
Differentiation in Mathematics refers to the process of finding the rate of change of a function with respect to one of its variables.
Differentiation is the process of finding the derivative of a function.
The product rule of differentiation states that if $f(x)$ and $g(x)$ are differentiable functions, then $(f(x)g(x))’ = f’(x)g(x) + f(x)g’(x)$.
Product Rule: $(\frac{d}{dx})(uv) = u(\frac{dv}{dx}) + v(\frac{du}{dx})$
The quotient rule of differentiation states that: ddx(f(x)g(x))=f′(x)g(x)−g′(x)f(x)[g(x)]2
(d/dx)(u/v) = (v (du/dx) - u (dv/dx))/v^2
What is the derivative of cot(x) with respect to x?
The derivative of cot x with respect to x = -cosec2x.
JEE Study Material (Mathematics)
- 3D Geometry
- Adjoint And Inverse Of A Matrix
- Angle Measurement
- Applications Of Derivatives
- Binomial Theorem
- Circles
- Complex Numbers
- Definite And Indefinite Integration
- Determinants
- Differential Equations
- Differentiation
- Differentiation And Integration Of Determinants
- Ellipse
- Functions And Its Types
- Hyperbola
- Integration
- Inverse Trigonometric Functions
- Limits Continuity And Differentiability
- Logarithm
- Matrices
- Matrix Operations
- Minors And Cofactors
- Properties Of Determinants
- Rank Of A Matrix
- Solving Linear Equations Using Matrix
- Standard Determinants
- Straight Lines
- System Of Linear Equations Using Determinants
- Trigonometry
- Types Of Matrices