Calculus is divided into two branches. One is differential calculus, another is integral calculus. Differential calculus deals with derivatives of functions and their applications. Derivative of a functions is defined as the rate of change of a function. Derivative of a function tells how the value of a function changes with any change in its input. The process of finding derivative is called differentiation. Differential calculus is widely used in physics also. |

$\frac{\mathrm{d}c }{\mathrm{d} x}$ = 0, where c a constant

$\frac{\mathrm{d} }{\mathrm{d} x}$$x^{n} = nx^{n - 1}$

$\frac{\mathrm{d} }{\mathrm{d} x}$$e^{x} = e^{x}$

$\frac{\mathrm{d} }{\mathrm{d} x}$$a^{x} = a^{x} \ln a$, for $a > 0$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\ln x$ = $\frac{1}{x}$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\sin x = \cos x$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\cos x = -\sin x$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\tan x = \sec ^{2}x$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\cot x = -\csc ^{2}x$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\sec x = \sec x \tan x$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\csc x = -\csc x \cot x$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\sin^{-1}x$ = $\frac{1}{\sqrt{1 - x^{2}}}$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\cos^{-1}x = -$$\frac{1}{\sqrt{1 - x^{2}}}$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\tan^{-1}x$ = $\frac{1}{1 + x^{2}}$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\cot ^{-1}x = -$$\frac{1}{1 + x^{2}}$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\sec ^{-1}x$ = $\frac{1}{x\sqrt{x^{2} - 1}}$

$\frac{\mathrm{d} }{\mathrm{d} x}$$\csc ^{-1}x = -$$\frac{1}{x\sqrt{x^{2} - 1}}$

Differentiation is a widely used concept. It is not limited to finding derivatives and solving problems. Along with mathematics, differentiation is applied to physics, chemistry, economics and many other fields. Most commonly used applications of differentiation are described below:

- Differentiation is used to find extrema, which includes maximum and minimum values of a function.
**For example:**Finding the maximum volume of a sphere that can fit into a square of given side. - Differentiation is used in finding the limit of a function. Sometimes, when we get indeterminate form, L'hospital rule is applied which is based on differentiation.
- In physics, various phenomena are based upon differentiation.
**For example:**Finding velocity, acceleration etc. with which most of the physics concepts are related. - Differentiation is widely used in economics in determining marginal cost, marginal profit etc.

A function can have more than one maximum (plural - maxima) or minimum (plural - minima) as shown in the figure:

### How to Find Maxima or Minima?

Following steps should be followed while finding local maximum and local minimum for a function ${f}(x)$:- Find first derivative of the function ${f}'(x)$.
- Equate it to zero and determine critical point (say "a"). Critical point is a point, where either maximum or minimum exists.
- Calculate second derivative ${f}''(x)$.
- Evaluate ${f}''(a)$
- If ${f}''(a) < 0$, then there is a maximum at point a.
- If ${f}''(a) > 0$, then there is a minimum at point a.

### Solved Example

**Question:**Find the maximum or minimum point for a function $f(x) = 3x^{2} - 2x - 1$.

**Solution:**

${f}'(x) = 6x - 2$

Now, let us find the critical point.

${f}'(x) = 0$

$6x - 2 = 0$

$x$ = $\frac{1}{3}$

Let us find the second derivative.

${f}''(x) = 6 > 0$

Therefore, there is a local minimum at point $x$ = $\frac{1}{3}$.

### Solved Examples

**Question 1:**Find the derivative of the function $y = x^{4} - 3x^{3} - 2\sqrt{x} + $$\frac{2}{7}$ with respect to x.

**Solution:**

$y = x^{4} - 3x^{3} - 2x$$^{\frac{1}{2}} + \frac{2}{7}$

$\frac{\mathrm{d} y}{\mathrm{d} x}$ = $4x^{3} - 3 \times 3 \times x^{2} - 2 \times $$\frac{1}{2}x^{-\frac{1}{2}}$$ + 0$

$\frac{\mathrm{d} y}{\mathrm{d} x}$ = $4x^{3} - 9x^{2} - $$\frac{1}{\sqrt{x}}$

**Question 2:**Find $\frac{\mathrm{d} f}{\mathrm{d} x}$, if $f(x) = \sec x - \tan x$.

**Solution:**

$\frac{\mathrm{d} f}{\mathrm{d} x}$ = $\tan x \sec x - \sec ^{2}x$

$ = \sec x(\tan x - \sec x)$