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# Differential Calculus

Differential calculus is very important part of math. It deals with real function, limits, continuity and differentiability, differentiation, tangents, normal and other application of derivatives etc.

Let us talk about real functions. The real function is a function which has domain and co-domain as the set of all real numbers. A real function f (x) is called continuous function, if it is continuous at each point in the domain. If we have a real function f (x), which have a point ‘a’ in the domain of a given function f (x) and if $\lim_{h\to 0}$ $\frac{f(a + h) - f (a)}{h}$ is defined, then the value of the given limit function at x = a is called the derivative of the function.

The process to find the derivative of the given function is called differentiation. The derivative of a function at any point x = a is the slope of the tangent to the curve y = f (x), at the point (a, f (a)).
Below are some standard derivative of functions.
1. Derivative of xn = nx(n - 1)
2. Derivative of ex = ex
3. Derivative of ax = ax log
4. Derivative of log (x) = $\frac{1}{x}$
Derivative of standard trigonometric functions are as follows:
1. Derivative of sin (x) = cos (x)
2. Derivative of cos (x) = - sin (x)
3. Derivative of tan (x) = sec2 (x)
4. Derivative of cot (x) = - cosec2 (x)
5. A derivative of sec (x) = sec (x) tan (x)
6. Derivative of cosec (x) = - cosec (x) cot (x)
Some of the fundamental rules of differentiation are as follows:
1. Derivative of a constant function is always zero.
2. Product rule of derivative: If we have two functions f (x) and g (x), then the derivative of f (x) x g (x) = f (x) x $\frac{d}{dx}$ g (x) + g (x) x $\frac{d}{dx}$ f (x).
Differential Calculus Examples:
Given below are some of the examples in differential calculus.

Example 1:

Find the derivative of f (x) = log ($x^{3}+2x^{2}+1$)

Solution:

Given:
f (x) = log ($x^{3}+2x^{2}+1$)

Differentiate both sides with respect to x.
F’(x) = derivative of log ($x^{3}+2x^{2}+1$)
= [1 / ($x^{3}+2x^{2}+1$)] x derivative of ($x^{3}+2x^{2}+1$)
= ($3x^{2}+4x$) / ($x^{3}+2x^{2}+1$)

Example 2:

Find the derivative of f (x) = log sin (x)

Solution:

Given: f (x) = log sin (x)

Differentiate both sides with respect to x.
f’(x) = $\frac{1}{\sin x}$ x derivative of sin (x)
= $\frac{\cos x}{\sin x}$ = cot (x)

Example 3:

Find the derivative of f (x) = $e^{x^{2}+3x +1}$

Solution:

Given: f (x) = $e^{x^{2}+3x +1}$

Differentiate both sides with respect to x.
f’(x) = derivative of $e^{x^{2}+3x +1}$
F’(x) = $e^{x^{2}+3x +1}$ x derivative of ${x^{2}+3x +1}$
f’(x) = (2x + 3) $e^{x^{2}+3x +1}$

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 More topics in  Differential Calculus Antiderivatives Slope Derivative Derivative Tangent and Normal
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