Derivative

Derivative rules

Derivative rules and laws. Derivatives of functions table.

Derivative definition
Derivative rules
Derivatives of functions table
Derivative examples

Derivative definition

The derivative of a function is the ratio of the difference of function value f(x) at points x+Δx and x with Δx, when Δx is infinitesimally small. The derivative is the function slope or slope of the tangent line at point x.

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

Second derivative

The second derivative is given by:

Or simply derive the first derivative:

Nth derivative

The nth derivative is calculated by deriving f(x) n times.

The nth derivative is equal to the derivative of the (n-1) derivative:

f⁽ⁿ⁾(x) = [f^(n-1)(x)]'

Example:

Find the fourth derivative of

f (x) = 2x⁵

f ⁽⁴⁾(x) = [2x⁵]'''' = [10x⁴]''' = [40x³]'' = [120x²]' = 240x

Derivative on graph of function

The derivative of a function is the slop of the tangential line.

Derivative rules

Derivative sum rule	( a f (x) + bg(x) ) ' = a f ' (x) + bg' (x)
Derivative product rule	( f (x) ∙ g(x) ) ' = f ' (x) g(x) + f (x) g' (x)
Derivative quotient rule	$\left ( \frac{f(x)}{g(x)} \right )'=\frac{f'(x)g(x)-f(x)g'(x)}{g^2(x)}$
Derivative chain rule	f ( g(x) ) ' = f ' ( g(x) ) ∙ g' (x)

Derivative sum rule

When a and b are constants.

( a f (x) + bg(x) ) ' = a f ' (x) + bg' (x)

Example:

Find the derivative of:

3x² + 4x.

According to the sum rule:

a = 3, b = 4

f(x) = x², g(x) = x

f ' (x) = 2x, g' (x) = 1

(3x² + 4x)' = 3⋅2x+4⋅1 = 6x + 4

Derivative product rule

( f (x) ∙ g(x) ) ' = f ' (x) g(x) + f (x) g' (x)

Derivative quotient rule

$\left ( \frac{f(x)}{g(x)} \right )'=\frac{f'(x)g(x)-f(x)g'(x)}{g^2(x)}$

Derivative chain rule

f ( g(x) ) ' = f ' ( g(x) ) ∙ g' (x)

This rule can be better understood with Lagrange's notation:

$\frac{df}{dx}=\frac{df}{dg}\cdot \frac{dg}{dx}$

Function linear approximation

For small Δx, we can get an approximation to f(x₀+Δx), when we know f(x₀) and f ' (x₀):

f (x₀+Δx) ≈ f (x₀) + f '(x₀)⋅Δx

Derivatives of functions table

Function name	Function	Derivative
Function name	f (x)	f '(x)
Constant	const	0
Linear	x	1
Power	x^a	a x^a-¹
Exponential	e^x	e^x
Exponential	a^x	a^xln a
Natural logarithm	ln(x)
Logarithm	log_b(x)
Sine	sin x	cos x
Cosine	cos x	-sin x
Tangent	tan x
Arcsine	arcsin x
Arccosine	arccos x
Arctangent	arctan x
Hyperbolic sine	sinh x	cosh x
Hyperbolic cosine	cosh x	sinh x
Hyperbolic tangent	tanh x
Inverse hyperbolic sine	sinh^-1 x
Inverse hyperbolic cosine	cosh^-1 x
Inverse hyperbolic tangent	tanh^-1 x

Derivative examples

Example #1

f (x) = x³+5x²+x+8

f ' (x) = 3x²+2⋅5x+1+0 = 3x²+10x+1

Example #2

f (x) = sin(3x²)

When applying the chain rule:

f ' (x) = cos(3x²) ⋅ [3x²]' = cos(3x²) ⋅ 6x

Second derivative test

When the first derivative of a function is zero at point x₀.

f '(x₀) = 0

Then the second derivative at point x₀ , f''(x₀), can indicate the type of that point:

f ''(x₀) > 0	local minimum
f ''(x₀) < 0	local maximum
f ''(x₀) = 0	undetermined

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