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Derivative (Differentiation) Calculator

To find the derivative enter function, select variable, write order, and press calculate button using derivative calculator

Table of Contents:

Derivative Calculator
What is a derivative?
Derivative Rules (Techniques)
Examples of derivatives
FAQs

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Derivative Calculator

Derivative calculator is used to calculate the derivative of a function. It is also known as the differentiation calculator because it solves a function by calculating its derivative for the variable.

What is a derivative?

A derivative is used to find the change in a function with respect to the change in a variable.

Britannica defines the derivatives as,

“In mathematics, a derivative is the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations.”

Wikipedia states that,

“The derivative of a function of a real variable measures the sensitivity to change of the output value with respect to a change in its input value.”

Derivative Rules (Techniques)

Below, you will find the basic and advanced derivative rules, which will help you understand the whole process of derivation.

Sum rule

$[f(x) + g(x)]' = f'(x) + g'(x)$

Constant rule

The derivative of any constant would be 0 in any case.

$f' (x) = 0$

Product rule

$[f(x) * g(x)]' = f'(x)g(x) + g'(x)f(x)$

If the above equation confuses you, use the product rule calculator above to differentiate a function using the product rule.

Quotient rule

$(f g)' = f'g - fg' g 2$

Chain rule

$If f(x) = h (g(x))$

$f'(x) = h' (g(x)).g' (x)$

This calculator also acts as a chain rule calculator because it uses the chain rule for derivation whenever it is necessary.

Derivatives cannot be evaluated by using a single static formula. There are specific rules to evaluate each type of function.

Exponent Rule	$d dx e x = e x$
Logarithmic Rules	$d dx a x = a x ln(a), a > 0$ $d dx ln(x) = 1 x, x > 0$ $d dx log x (x) = 1 x ln(a), x , x > 0$
Trigonometric Rules	$d dx sin(x) = cos(x)$ $d dx cos(x) = -sin(x)$ $d dx tan(x) = sec 2 (x) = 1 cos 2 (x) = 1 + tan 2 (x)$
Inverse Trigonometric Rules	$d dx arcsin(x) = 1 1 - x 2$ $d dx arccos(x) = - 1 1 - x 2$ $d dx arctan(x) = 1 1 - x 2$
Power Rule	$d dx x a = ax (a-1)$

Examples of derivatives

Example 1

Find out the derivative of the following function.

$f(x) = (x 2 + 5) 3$

Solution:

Step 1: As we can see, the given function can be evaluated by chain rule.

$f(x) = (x 2 + 5) 3$

Step 2: Write down the chain rule.

$f'(x) = h'(g(x)).g' (x)$

Step 3: Let’s apply the chain rule to the given function.

$f'(x) = 3(x 2 + 5) 3-1 f'(x 2 + 5)$

The left part of the function is evaluated. Now, to solve the right part of the function, we can apply the sum rule because the expression contains the sum operator.

$f'(x) = 3(x 2 + 5) 2 (f'(x 2) + f'(5))$

$f'(x) = 3(x 2 + 5) 2 ((2x) + (0)) \to f'(x) = 0$

$f'(x) = 6x(x 2 + 5)$

Example 2

Solve the derivative of the given function.

$f(x) = (x 3 - 2)(x 2 + x - 4)$

Solution:

Step 1: Here, we will use the product rule to solve the given expression.

$f(x) = (x 3 - 2)(x 2 + x - 4)$

Step 2: Write down the product rule.

$(fg)' = f'g + fg'$

Step 3: Apply the product rule to solve the expression.

$f'(x) = (x 2 + x - 4) f'(x 3 - 2) f'(x 2 + x -4)$

$f'(x) = (x 2 + x - 4) f'(x 3) f'(2)) + (x 3 - 2) (f'(x 2) + f'(x 2) + f'(x) -f'(4))$

$f'(x) = (x 2 + x - 4) (3x 2 - 0) + (x 3 - 2 ) (2x + 1 - 0)$

$f'(x) = 3x 2 (x 2 + x - 4) + (x 3 - 2) (2x + 2 )$

FAQs

How do you calculate derivatives?

Derivatives can be calculated in several ways according to the function. The derivative of a constant would be zero. There are numerous rules of derivation which we can apply according to the nature of the function, i.e., sum, product, chain rule, etc.

$f(x) = x 2 + 2x - 3$

$f'(x) = 2x 2-1 + 2(1) - 0$

$f'(x) = 2x + 2$

Why do we calculate derivatives?

We calculate the derivatives to compute the rate of change in one object because of the change in another object. For example, $dx dy$ simply means that we are calculating the total change that occurred in x object due to the change in y object.

What's a derivative in math?

In mathematics, a derivative is the measure of the rate of change with respect to a variable. For example, we can calculate the change in the speed of a car for a specific time period using time as a variable.