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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,
Derivative Formula 

Related Calculators
Derivative Calculator Anti Derivative Calculator
 

Derivative Formula List

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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

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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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