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# Derivative Formula

Derivative is defined as the slope or rate of change of the tangent line. Derivative of a function is the measure of the changes in the function with respect to the given variable. This process is also known as differentiation and on in-versing this process we obtain the anti derivative.

Derivative Formula is given as,

 Related Calculators Derivative Calculator Anti Derivative Calculator

## Derivative Formula List

Below is the list of derivative formals:

Derivative of a constant :
$\frac{\mathrm{d} (k)}{\mathrm{d} x}$ = 0

Derivative of a constant times a function :
$\frac{\mathrm{d} (k(u(x)))}{\mathrm{d} x}$$k \frac{\mathrm{d}(u) }{\mathrm{d} x}$

Derivative of a variable raised to a constant :
$\frac{\mathrm{d} (u^{n})}{\mathrm{d} x}$ = $n(u)^{n-1}\frac{\mathrm{d} (u))}{\mathrm{d} x}$

Derivative sum rule :
$\frac{\mathrm{d}(u + v)}{\mathrm{d} x}$ = $\frac{\mathrm{d} (u))}{\mathrm{d} x}$ + $\frac{\mathrm{d} (v)}{\mathrm{d} x}$

Derivative difference rule :
$\frac{\mathrm{d}(u - v)}{\mathrm{d} x}$ = $\frac{\mathrm{d} (u))}{\mathrm{d} x}$ - $\frac{\mathrm{d} (v)}{\mathrm{d} x}$

Derivative product rule :
$\frac{\mathrm{d}(uv)}{\mathrm{d} x}$ = uv' + vu'

Derivative quotient rule :
$\frac{\mathrm{d}}{\mathrm{d} x}\left(\frac{u}{v}\right )$ = $\frac{vu' - uv'}{v^{2}}$

Derivative chain rule :
$\frac{\mathrm{d}y}{\mathrm{d} x}$ = $\frac{\mathrm{d}y}{\mathrm{d}u}$ $\times$ $\frac{\mathrm{d}u}{\mathrm{d}x}$

## Derivative Problems

Some solved problems on derivative are given below:

### Solved Examples

Question 1: Calculate the derivative of f(x) = 5x2 + 2 ?
Solution:

Given,
f(x) = 5x2 + 2

The derivative formula is,
f'(x) = $\lim_{\Delta x \to 0}$  $\frac{f(x + \Delta{x})-f(x)}{ \Delta{x}}$

f'(x) = $\lim_{\Delta x \to 0}$  $\frac{[5(x+ \Delta{x})^{2}+2] - [5x^{2}+2] }{\Delta{x}}$

f'(x) = $\lim_{\Delta x \to 0}$  $\frac{[5(x^{2} + 2x \Delta{x} +\Delta x^{2}) + 2] - [5x^{2} + 2] }{ \Delta{x}}$

f'(x) = $\lim_{\Delta x \to 0}$  $\frac{[5x^{2} + 10x \Delta{x} +5 \Delta x^{2} + 2] - [5x^{2} + 2] }{ \Delta{x}}$

f'(x) = $\lim_{\Delta x \to 0}$  $\frac{10x \Delta{x} +5 \Delta x^{2} }{ \Delta{x}}$

f'(x) = $\lim_{\Delta x \to 0}$  $\frac{\Delta{x} (10x + 5 \Delta x)}{ \Delta x}$

f'(x) = $\lim_{ \Delta x \rightarrow 0} 10x + 5 \Delta x$

f'(x) = 10x

Question 2: Calculate the derivative of f(x) = 2x ?
Solution:

Given,
f(x) = 2x

The derivative formula is,
f'(x) = $\lim_{\Delta x \to 0}$  $\frac{f(x+\Delta x)-f(x)}{ \Delta x}$

f'(x) = $\lim_{\Delta x \to 0}$ $\frac{2(x+\Delta x)-2x}{ \Delta x}$

f'(x) = $\lim_{\Delta x \to 0}$ $\frac{2x + 2 \Delta x - 2x}{ \Delta x}$

f'(x) = $\lim_{\Delta x \to 0}$ $\frac{2 \Delta x}{ \Delta x}$

f'(x) = $\lim_{\Delta x \to 0}$ 2

f'(x) = 2

 More topics in Derivative Formula Antiderivative Formula Chain Rule Formula Instantaneous Rate of Change Formula Product Rule Formula Implicit Differentiation Formula Mean Value Theorem Formula Quotient Rule Formula Newton's Method Formula
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