Derivative rules

Derivative rules and laws. Derivatives of functions table.

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.

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)]'

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
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)

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
	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 x ln a
Natural logarithm	ln(x)
Logarithm	logb(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

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:

Derivatives are a cornerstone in calculus, providing a powerful tool for understanding how a function changes. Calculating derivatives involves finding the rate at which a function changes with respect to its independent variable. Whether you're a student diving into calculus or a professional using mathematical tools, understanding derivatives is crucial.

This fundamental concept finds applications in physics, economics, engineering, and more. Whether analyzing the motion of objects, modeling financial trends, or optimizing processes, derivatives play a key role.

Calculating derivatives involves various techniques, such as the power rule, product rule, quotient rule, and chain rule. Our guide breaks down these methods, offering step-by-step explanations and examples to facilitate a clear understanding. Dive into the world of derivatives with our comprehensive guide and unlock their potential in mathematical problem-solving.