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# Derivative (Differentiation) Calculator

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

## Differentiation calculator

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.

ddx (3x + 92 - 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 content, 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 differentiatial 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.

- 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:

dydx = dfdx

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 Rules (Techniques)

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

( fg )^{'} = 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

ddxx^{a} = ax^{(a-1)}

#### · Exponents

For the **derivative of e ^{x},**

ddxe^{x} = e^{x}

#### · Logarithmic functions

ddx a^{x} = a^{x} ln(a), a > 0

ddx ln(x) = 1x , x > 0

ddx log_{x}(x) = 1x ln(a) , x , x > 0

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

#### · Trigonometric functions

ddx sin(x) = cos(x)

ddx cos(x) = -sin(x)

ddx tan(x) = sec^{2}(x) = 1cos^{2}(x) = 1 + tan^{2}(x)

#### · Inverse trigonometric functions

ddx arcsin(x) = 11 - x^{2}

ddx arccos(x) = - 11 - x^{2}

ddx arctan(x) = 11 - x^{2}

If you want to find the Second derivative then use our second devrivative calculator.

## 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)) → 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 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, dxdysimply means that we are calculating the total change that occurred in ** x **object due to the change in

**object.**

*y*### 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 investopedia.com.
- Calculus I - The Definition of the Derivative by tutorial.math.lamar.edu.