 # Function derivatives examples

Know how and good skills in functions derivatives calculation is very important in science and engineering. On this website you can find plenty of examples about functions derivatives which are solved step by step. We often use functions derivatives to find maxium or minimum value of function. Good mathematical skills are very important.

## Basic functions derivatives We are lucky because we don't have to designate all derivatives of functions. Mathematicians had designated plenty of basic derivatives of functions. Most of functions which you will meet are complex of basic functions. Below are placed basic functions derivatives and some basic theorems about function derivatives.

Basic derivatives of functions

## Function derivative - example 1 Calculation of function derivative which form is f(x)=2·x. Definition of derivative will be used to calculate derivative of this function.

Function derivative - example 1

## Function derivative - example 2 Calculation of function derivative which form is f(x)=x2. Definition of derivative will be used to calculate derivative of this function.

Function derivative - example 2

## Function derivative - example 3 Calculation of function derivative which form is f(x)=3·x2+4·x-1.

Function derivative - example 3

## Function derivative - example 4 Calculation of function derivative which form is f(x)=(x-5)·ex.

Function derivative - example 4

## Function derivative - example 5 Calculation of function derivative which form is f(x)=cos(5·x)+e-2·x.

Function derivative - example 5

## Function derivative - example 6 Calculation of function derivative which form is f(x)=cos(5·x)·e-2·x. Considered function is multiply of two complex functions.

Function derivative - example 6

## Function derivative - example 7 Calculation of function derivative which form is f(x)=(3/(x+7))2. Considered function is a complex functions.

Function derivative - example 7

## Function derivative - example 8 Calculation of function derivative which form is f(x)=sin2x. Considered function is a complex functions.

Function derivative - example 8

## Function derivative - example 9 Calculation of function derivative which form is f(x)=2·x·e-x2. Considered function is a multipy of complex function and non-complex function.

Function derivative - example 9

## Function derivative - example 10 Derivative calculation of function f(x)=x1/2+ln(x). Considered function is a sum of two component functions. First component function is a square root of variable x. Second component function is natural logarithm of variable x.

Function derivative - example 10

## Function derivative - example 11 Derivative calculation of function f(x)=x1/2·ln(x). Considered function is a multiply of two component functions. First component function is a square root of variable x. Second component function is natural logarithm of variable x. Theorem about derivative of two functions multiply will be applied.

Function derivative - example 11

## Function derivative - example 12 Derivative calculation of function f(x)=x4·ex. Considered function is a multiply of two component functions. First component variable x taken to the fourth power. Second component function is constant e to the x power. Theorem about derivative of two functions multiply will be applied.

Function derivative - example 12

## Function derivative - example 13 Derivative calculation of function f(x)=x/(x2+1). Considered function is a divide of two component functions. First component variable x taken to the first power. Second component function is constant one plus variable x taken to the second power. Theorem about derivative of two functions multiply will be applied. Theorem about derivative of complex function will be applied.

Function derivative - example 13

## Function derivative - example 14 Derivative calculation of function f(x)=x1/2. Considered function is a square root of variable x. On the other hand this function may be written in form as variable x taken to the power of one-half.

Function derivative - example 14

## Function derivative - example 15 Derivative calculation of function f(x)=(x1/2+1)/(x1/3). Considered function is a divide of two component functions. First component function is a constant one plus variable x in power one-half. Second function is variable x in power one-third.

Function derivative - example 15

## Function derivative - example 16 Derivative calculation of function f(x)=(sin(x))/(x+1). Considered function is a divide of two component functions. First component function is function sin(x). Second function is variable x plus one.

Function derivative - example 16

## Function derivative - example 17 Derivative calculation of function f(x)=x/cos(x). Considered function is a divide of two component functions. First component function is variable x. Second function is cos(x). Theorems about derivative of two function multiply and derivative of complex function will be applied.

Function derivative - example 17

## Function derivative - example 18 Derivative calculation of function f(x)=(1-x3)/(1+x3). Considered function is a divide of two component functions. First component function is (1-x3). Second component function is (1+x3). Theorems about derivative of two function multiply and derivative of complex function will be applied.

Function derivative - example 18

## Function derivative - example 19 Derivative calculation of function f(x)=x·sin2x. Considered function is a multiply of two component functions. First component function is x. Second component function is sin2x. Theorems about derivative of two function multiply and derivative of complex function will be applied.

Function derivative - example 19

## Function derivative - example 20 Derivative calculation of function f(x)=x·(1+x2)1/2. Considered function is a multiply of two component functions. First component function is x. Second component function is (1+x2)1/2. Theorems about derivative of two function multiply and derivative of complex function will be applied.

Function derivative - example 20

## Function derivative - example 21 Derivative calculation of function f(x)=e(1+x)/(1-x). Considered function is a complex function. Theorems about derivative of two function multiply and derivative of complex function will be applied.

Function derivative - example 22

## Function derivative - example 22 Derivative calculation of function f(x)=x2·e2·x. Considered function is a multiply of two component functions. First component function is x2. Second component function is e2·x. Theorems about derivative of two function multiply and derivative of complex function will be applied.

Function derivative - example 22

## Function derivative - example 23 Derivative calculation of function f(x)=(1+x2)1/2·ln(x). Considered function is a multiply of two component functions. First component function is (1+x2)1/2. Second component function is ln(x). Theorems about derivative of two function multiply and derivative of complex function will be applied

Function derivative - example 23

## Function derivative - example 24

Derivative calculation of function f(x)=sin(ln(2·x)). Considered function is a complex function. Outside function is sin(). First inner function is ln(). second inner function is 2·x. Theorems about derivative of two function multiply and derivative of complex function will be applied.

Function derivative - example 24

## Function derivative - example 25

Derivative calculation of function f(x)=sin2(2·x). Considered function is a complex function. Outside function is sin2(). Inner function is 2·x. Theorems about derivative of two function multiply and derivative of complex function will be applied.

Function derivative - example 25

## Function derivative - example 26

Derivative calculation of function f(x)=(1+ln(x2+1))1/2. Considered function is a triple complex function. Outside function is power(1/2). First inner function is natural logarithm ln(). Second inner function is x2+1. Theorems about derivative of complex function will be applied.

Function derivative - example 26

## Function derivative - example 27

Derivative calculation of function f(x)=sinh(x)=(ex-e-x)/2. It is know from formulas for derivatives of elementary functions that derivative of hiperbolic sine is a hiperbolic cosine. In example derivative of hiperbolic sine is calculated for its exponent form.

Function derivative - example 27

## Function derivative - example 28

Derivative calculation of function f(x)=cosh(x)=(ex+e-x)/2. It is know from formulas for derivatives of elementary functions that derivative of hiperbolic cosine is a hiperbolic sine. In example derivative of hiperbolic cosine is calculated for its exponent form.

Function derivative - example 28

## Function derivative - example 29

Derivative calculation of function f(x)=5·x3+2·x4+sin(x). Considered function is a sum of three component function. Theorem about derivative for sum of three functions will be used.

Function derivative - example 29

## Function derivative - example 30

Derivative calculation of function f(x)=(sin(x))3+(cos(x))3. Considered function is a sum of two component functions. Both component functions are complex functions. Theorems about derivative of sum of two functions and derivative of complex function will be used.

Function derivative - example 30

## Function derivative - example 31

Derivative calculation of function f(x)=(sin(x))3·(cos(x))3. Considered function is a multiply of two component functions. Both component functions are complex functions. Theorems about derivative of multiply of two functions and derivative of complex function will be used.

Function derivative - example 31