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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. This differentiation calculator differentiates the function step by step.

What is a derivative?

A derivative is used to find the change in a function with respect to the change in a variable. “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 & Difference Rule

According to the sum & difference rule the notation of differentiation will be applied to each term of the function separately.

d/dx [f(x) ± g(x)] = d/dx [f(x)] ± d/dx [g(x)]

Constant rule

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

d/dx K = 0

Where K is any constant

Product rule

d/dx [f(x) * g(x)] = d/dx [f(x)] * g(x) + f(x) d/dx [g(x)]

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

Quotient rule

d/dx [f(x) / g(x)] = 1/[g(x)]2 [d/dx [f(x)] * g(x) + f(x) d/dx [g(x)]]

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.

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

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

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

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.

$d/dx [f(x) * g(x)] = d/dx [f(x)] * g(x) + f(x) d/dx [g(x)]$

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

$d/dx f(x) = (x 2 + x - 4) d/dx [x 3 - 2] + (x 3 - 2) d/dx [x 2 + x - 4]$

$Differentiate the above expression with the help of sum, difference, constant, and power rules.$

$d/dx f(x) = (x 2 + x - 4) [d/dx(x 3) - d/dx(2)] + (x 3 - 2) [d/dx(x 2) + d/dx(x) - d/dx(4)]$

$d/dx f(x) = (x 2 + x - 4) [3x 2 - 0] + (x 3 - 2) [2x + 1 - 0]$

$d/dx f(x) = (x 2 + x - 4) [3x 2] + (x 3 - 2) [2x + 1]$

$d/dx f(x) = (3x 4 + 3x 3 - 12x 2) + (2x 4 + x 3 - 4x - 2)$

$d/dx f(x) = 5x 4 + 4x 3 - 12x 2 - 4x - 2$

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.