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Derivative Calculator

Table of Contents:

How to use derivative calculator?
What is a derivative?
Derivative formula
How to calculate derivative?
FAQs

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Derivative calculator can be 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.

$d dx (3x + 9 2 - x) = 15 (2 - x) 2$

Most students find it difficult to understand the concepts of differentiation because of the complexity involved. There are several types of functions in mathematics, i.e., constant, linear, polynomial, etc. This differential calculator can recognize each type of function to find the derivative.

In this article, we will explain the rules of differentiation, how to find derivative, how to find the derivative of the function such as derivative of x or derivative of 1/x, derivative definition, the formula of derivative, and some examples to clarify the calculations of differentiation.

How to use derivative calculator?

You can use the differentiate calculator to perform a differentiation on any function. The above implicit differentiation calculator proficiently parses the given function to place any missing operators in the function. Then, it applies the relative differentiation rule to conclude the result.

To use derivatives calculator,

Enter the function in the given input box.
Press the Calculate
Use the Reset button to enter a new value.

You can use this derivative calculator with steps to understand the step by step calculation of the given function. Moreover, you can also calculate the reverse derivative of a function by using our integral calculator.

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.”

After taking the first derivative of a function $y = f (x)$ it can be written as:

$dy dx = df dx$

If there is more than one variable involved in a function, we can perform the partial derivation by using one of those variables. Partial derivation can also be calculated using the partial derivative calculator above.

Derivative formula

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

Sum rule

$(af + βg)' = af' + βg'$

Constant rule

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

$f' (x) = 0$

Product rule

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

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.

Derivative of:

· Powers

$d dx x a = ax (a-1)$

· Exponents

For the derivative of $e x$ ,

$d dx e x = e x$

· Logarithmic functions

$d dx a x = a x ln(a), a > 0$

$d dx ln(x) = 1 x, x > 0$

$d dx log x (x) = 1 x ln(a), x , x > 0$

Logarithmic differentiation calculator effortlessly implements these rules to the given expressions.

· Trigonometric functions

$d dx sin(x) = cos(x)$

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

$d dx tan(x) = sec 2 (x) = 1 cos 2 (x) = 1 + tan 2 (x)$

· Inverse trigonometric functions

$d dx arcsin(x) = 1 1 - x 2$

$d dx arccos(x) = - 1 1 - x 2$

$d dx arctan(x) = 1 1 - x 2$

As a second derivative calculator, this tool can also be used to find the second derivative as well as the derivative of square root.

How to calculate derivative?

It is very convenient to find the derivative of any function using the derivative finder tool, but, it is recommended that you should go through basic concepts to master the topic.

In this space, we will explore step by step method to calculate derivatives. Here are the steps to find the derivative without using a derivative solver.

Write down the function and simplify it if needed.
Identify the type of function and write down the related rule.
Use the applicable rule from above to solve the function.

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.

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

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

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

How do you find the derivative quickly?

Use the implicit derivative calculator above to quickly find the derivative of a function or algebraic expression. You will get the result of differentiation in a few seconds.

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.

References:

Derivative -- from Wolfram MathWorld.
Calculus I - The Definition of the Derivative by tutorial.math.lamar.edu.