Derivative Calculator

The derivative calculator is a free online tool that displays the derivative of the given function. BYJU’S online derivative calculator or differentiation calculator tool makes the calculations faster, and it shows the first, second, third-order derivatives of the function in a fraction of seconds.

How to Use the Derivative Calculator?

The procedure to use the derivative calculator (differentiation calculator) is as follows:

Step 1: Enter the function in the respective input field and choose the order of derivative

Step 2: Now click the button “Calculate” to get the derivative

Step 3: The derivative of the given function will be displayed in the new window

What is the Derivative of a Function?

In calculus, one of the basic concepts is the derivative of a function. It occupies the central concept in calculus. We know that differentiation and integration are the two important concepts. Differentiation is the process of finding the derivative of a function, whereas integration is the process of finding the antiderivative of a function. The derivative of a function describes the rate of change. That means that it shows the amount by which the function is changing at the given point.

Standard Form

The standard form to represent the derivative of a function is given below:

An infinitesimal change in the variable “x” is denoted by dx. Thus, the derivative of the variable “y” with respect to the variable “x” is given by dy/dx.

Solved Examples on Derivatives

Example 1:

Find the first derivative of f(x) = 8x² + 12x.

Solution:

Given function: f(x) = 8x² + 12x.

Now, differentiating the function with respect of x, we get

(d/dx) (8x² + 12x) = (d/dx) (8x² ) + (d/dx)(12x)

(d/dx) (8x² + 12x) = 16x + 12

Therefore, the first order derivative of the function 8x² + 12x is 16x + 12.

Example 2:

Find the third derivative of f(x) = 14x⁴ – 2x.

Solution:

Given function: f(x) = 14x⁴ – 2x

Now, differentiate the function with respect to x, we get

First derivative:

(d/dx)(14x⁴ – 2x) = (d/dx)(14x⁴) – (d/dx)(2x)

(d/dx)(14x⁴ – 2x) = 56x³ – 2

Second derivative:

(d²/dx²)(14x⁴ – 2x) = 168x² – 0

Third derivative:

(d³/dx³)(14x⁴ – 2x) = 336x. 

Therefore, the third derivative of f(x) = 14x⁴ – 2x is 336x.

Example 3:

Find the fifth derivative of f(x) = 6x⁷ + 5x³ – 2x.

Solution:

Given function: f(x) = 6x⁷ + 5x³ – 2x.

To find the derivatives of the given function, differentiate the function with respect to x.

First derivative:

(d/dx)(6x⁷ + 5x³ – 2x) = (d/dx)(6x⁷) + (d/dx)(5x³) – (d/dx)(2x)

(d/dx)(6x⁷ + 5x³ – 2x) = 42x⁶ + 15x² – 2

Second derivative:

(d²/dx²)(6x⁷ + 5x³ – 2x) = 252x⁵ + 30x – 0

Third derivative:

(d³/dx³)(6x⁷ + 5x³ – 2x) = 1260x⁴  + 30

Fourth derivative:

(d⁴/dx⁴)(6x⁷ + 5x³ – 2x) = 5040x³ + 0

Fifth Derivative:

(d⁵/dx⁵)(6x⁷ + 5x³ – 2x) = 15120x²

Hence, the fifth derivative of f(x) = 6x⁷ + 5x³ – 2x is 15120x².

Also, check out: Implict Differentiation Calculator.