icon-Mathe Derivative Calculator and Derivative Calculus

Derivatives

Derivatives of basic functions.

d d x Const. = 0

d d x x = 1

d d x xn = nxn-1

Derivative of trigonometric functions

d d x sin(x) = cos(x)

d d x cos(x) = -sin(x)

d d x sin(kx) = kcos(kx)

d d x cos(kx) = -ksin(kx)

d d x tan(x) = d d x sin(x) cos(x) = 1 cos2(x)

Derivation exponential and logarithmic functions

d d x ex = ex

d d x ln(x) = 1x

d d x loga(x) = 1xloga(e)

Derivative Calculator

The derivative calculator calculates the first and second derivative of a function f(x).

f(x) =

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a⋅sin(b⋅x+c)

a⋅e(-b⋅x2+c)

a⋅x2+b⋅x+c

e(x)⋅sin(x)⋅cos(x)

Differential Calculus

Derivative calculus rules

The following are the most important differentiation rules described and illustrated by examples.

  • Factor Rule
  • Sum rule
  • Product Rule
  • Quotient Rule
  • Chain Rule

Factor rule and Sum rule

Ableitungsregel-1

The sum rule states that the summands can be individually differentiated.

d d x f1x + f2x = d d x f1x + d d x f2x

Derivation of the summands

The factor rule states that the constant factors are conserved during derivation.

d d x a fx = a d d x fx

The constant factor a is retained when deriving

Example of factor and sum rule

fx = 2x2+ 2 3 x3

The example function contains sum and constant factors. To differentiate, both rules are applied.

d d x 2x2 + 2 3 x3 = d d x 2x2 + d d x 2 3 x3

Application of the sum rule.

d d x 2x2 + d d x 2 3 x3 = 2 d d x x2 + 2 3 d d x x3

Apply the factor rule in each part of the sum.

2 d d x x2 + 2 3 d d x x3 =4x+2x2

Deriving the terms gives the derivation of the example function f.

Product rule

Ableitungsregel-2

The product rule specifies how to handle the product of two functions when differentiating. In words, the product rule can be expressed as follows: Derivation of the first function times the second function plus the first function times derivation of the second function.

Examples for Product rule

Ableitungsregel-1

If a product consists of more than two functions, then the product rule can be used successively by combining functions as required and applying the product rule several times in succession.

Ableitungsregel-1

Quotient rule

Ableitungsregel-3

The quotient rule specifies how to treat the quotient of two functions when differentiating.

Example for Quotient rule

Ableitungsregel-1

Chain rule

Ableitungsregel-4

The chain rule specifies how nested functions are to be treated when differentiating. One distinguishes between the inner function and the outer function. Thus, the chain rule can be formulated as follows: the derivative is derivative of the inner function times the derivative of the outer function. In the derivation of the outer function, the inner function as a whole is considered as variable. That it is not differentiated by x but by the inner function g.

Example for Chain rule

Ableitungsregel-1

Mixed differentiations

The following examples show the mixed use of the derivation rules.

Ableitungsregel-4 Ableitungsregel-4 Ableitungsregel-4