Derivative Calculator

Want to find the derivation of a function? Get accurate and step by step solution using our differentiation/derivative calculator with steps.


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Introduction to Derivative Calculator

Derivative calculator with steps helps users to compute first, second, or even nth derivative functions with many variables. With that, you can check your answer in the form of a graph for better understanding.

In calculus, derivatives are a fundamental concept used to describe the rate at which a function changes with respect to its independent variable. They have widespread applications in various fields such as physics, engineering, economics, and more. However, computing derivatives by hand can be time-consuming and error-prone, especially for complex functions.

derivative calculator

Our calculator also computes the derivative of a given function with respect to its independent variable. It is primarily used to find the rate of change of a function, identify critical points, determine slopes of tangents, and analyze the behavior of functions.

Additionally, if you would like to computes the second derivative of a function, which represents the rate of change of the first derivative, you can utilize our second derivative calculator. Our calculator also provides information about the curvature, concavity, inflection points, and the rate of change of the rate of change of the original function.

What are Derivatives?

Differentiation can be defined as a small rate of change in one quantity (dependent variable) with respect to another quantity (independent variable). The changing relationship between two variables can be determined by using differentiation.

Derivatives have broad applications across mathematics, science, engineering, economics, and various other fields. They are foundational not only in calculus but also in understanding the behavior and relationships of functions in real-world scenarios.

Further, the changing relationship between two variables can be determined by using differentiation. If you would like to detremind the relationship between triple variable, you should go through our third derivative calculator. This calculator also help you to find the solution to the rate of change of acceleration problem in the run of time.

Notion use for derivation

There are different types of notions used for differential

Derivative of a function f(x) with respect to x can be denoted as:

$$ \frac{dy}{dx} \; or \; \frac{d}{dx} (f(x)) \;\;\;\;\;\;\; Leibniz \; notion $$

where y is an dependent value and x is independent variable

$$ y' or f'(x) \;=\; \;\;\;\;\;\;\; Lagrange \; or \; Prime \; notion $$ $$ Df(x)\;=\; \frac{dy}{dx} \;\; as \;\; y\;=\;f(x) \;\;\;\;\;\;\; Euler notion $$

you can use any notion which mention above to represent differentiation

Derivation of a function f(x) with respect to x represent at the rate of change of function f.

$$ f'(x) \;=\; \lim _{h \to 0} \frac{f(x+h) - f(x)}{h} $$

Importance of Using Differentiation Calculator

Derivative solver is an online tool that helps to differentiate a function. It provides tons of solutions even if complex input is given by a user. This online calculator has done faster calculations than man.

In mathematical calculus, derivation is a vast topic. You can solve simple differentiation easily by using rules of derivation which are mentioned below. But when we go into solving higher-order derivatives it would be difficult for the human mind to calculate such complex problems.

But no need to worry because our differential calculator has all built-in formulas like exponential, logarithm, and trigonometry functions, which are used for solving complex problems. You can solve these complicated differential equations for projects, assignments, or reports by using our differentiate calculator for accurate calculations.

Our calculators also play a crucial role in mathematical analysis and problem-solving by offering efficiency, accuracy, versatility, visualization, and educational benefits.

Moreover, if you would like to determine the differentiation of multivariable independent functions in a run of time, You can utilize our partial derivative calculator. This calculator also to explore partial derivatives, students can reinforce their understanding of differentiation concepts in multiple dimensions and gain practical experience with calculus principles.

Rule Behind Our Derivative Calculator

Derivatives calculator uses all forms of derivative rules like product rule ,chain rule, quotient rule etc, so you solve a variety of problems.

Constant Rule

The derivative of a constant is 0.

f'(c) = 0

Power Rule

The derivative of a power rule is:

$$ \frac{d}{dx} x^n \;=\; nx^{n-1} $$

Sum Rule

If f(x) and g(x) are two differentiable functions with respect to x, then the derivative of their sum. Mathematically,

$$ \frac{d}{dx} (f(x) + g(x)) \;=\; \frac{d}{dx} f(x) + \frac{d}{dx} g(x) $$

Product Rule

If you have two functions i.e. f(x) and g(x), then product rule stated as:

$$ (f \cdot g)' \;=\; f' \cdot g + f \cdot g' $$

Quotient Rule

If you have two functions i.e. f(x) and g(x), then quotient rule stated as:

$$ f'(x) \;=\; \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2} $$

Chain Rule

Mathematically, the chain rule is expressed as:

$$ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} $$

A derivative calculator with steps is an effective tool for getting solutions for multiple functions by changing its values. has wide applications in mathematics and science.

Related: for calculations involving directional derivatives, you can use our directional derivative calculator. Our calculator also helps to evaluate the rate of change of the function at a particular point in a fixed direction within a minute.

How to Use Our Derivative Solver?

Our differentiation calculator is an online tool designed to calculate the derivative of a given function in the simplest method. In the derivative calculator, when its input changes the differential calculator shows how the value of a function changes.

  1. Enter a function you would like to differentiate. You should be cautious of the syntax used for mathematical problems.
  2. Enter your desired differentiation variable, if it is different from the default value.
  3. Choose an order of differentiation.
  4. Click the ‘Calculate’ button.
  5. After computing the derivative function your answer will show on the screen.
  6. Click on the ‘ a step-by-step solution’ button if you would like to see the differentiation steps.
  7. Click on “Show graph” to display graphs of the function on screen and its derivatives for better understanding.

How to Evaluate Derivatives Online?

Derivative solver will calculate the derivative of any function using the common rules of differentiation (product rule, quotient rule, chain rule, etc.), with a graph display. It can also calculate polynomial, rational, irrational, exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions. Also, it will evaluate the derivative of complex functions if needed.

Solved Examples

Let’s understand how our online differentiation calculator evaluates results with step by step solutions.

Example 1: Find the derivative of 5x3 + 2x2.


$$ f(x) \;=\; 5x^3 + 2x^2 $$ $$ f'(x) \;=\; \frac{d}{dx} (5x^3 + 2x^2) $$ $$ f'(x) \;=\; \frac{d}{dx} (5x^3) + \frac{d}{dx} (2x^2) $$ $$ f'(x) \;=\; 5 . \frac{d}{dx} (x^3) + 2 . \frac{d}{dx} (x^2) $$ $$ f'(x) \;=\; 5(3x^2) + 2(2x) $$ $$ f'(x) \;=\; 15x^2 + 4x $$

Thus it is the final solution of our function with specific limits. Further, if you want to find the nth derivative of this expression, you can use our nth derivative calculator. You would apply the power rule iteratively for each derivative until you reach the nth derivative.

Example 2: Find the derivative 13x2 + 8.


$$ f(x) \;=\; 13x^2 + 8 $$ $$ f'(x) \;=\; \frac{d}{dx} (13x^2 + 8) $$ $$ f'(x) \;=\; \frac{d}{dx} (13x^2) + \frac{d}{dx}(8) $$ $$ f'(x) \;=\; 13 . \frac{d}{dx} (x^2) + \frac{d}{dx}(8) $$ $$ f'(x) \;=\; 13 (2x) + 0 $$ $$ f'(x) = 26x $$

Similarly, you can use the derivative calculator with steps to find the derivatives of the following functions:

$$ \frac{d}{dx} \frac{x^3}{2} $$ $$ \frac{d}{dx} 5x^2 + 6x^7 $$

Thus it is the final solution of our function with specific limits. Additionally if you would like to solve the fourth derivative of the result, you should use our fourth derivative calculator. Our calculators can swiftly compute derivatives of functions up to the desired order, saving time and effort.

Result of Differential Calculator

Differentiate calculator computes derivatives of a function with respect to given variables using analytical differentiation and displays a step-by-step solution. It allows drawing graphs of the function and its derivation to give a better understanding to the reader.

Derivatives calculator supports higher-order derivatives as well as complex function differentiation. Derivatives are computed by parsing the function using correct mathematical syntax and getting results into the simplest form.

Use our calculator to get a step-by-step format that others cannot provide. You can check your answer manually to check the accuracy of our differentiation calculator.

Related: For a comprehensive understanding of function behavior, you can utilize our using our inflection point calculator to pinpoint inflection points and understand how functions curve.

Why to Choose this Differentiate Calculator?

In our derivative calculator all rules are installed in its programming which means you can use it for any type of function like trigonometry, exponential, or for higher order derivatives.

  1. It provides accurate calculations while differentiating any complex function.
  2. Our user-friendly interface give the user the simplest procedure for calculation.
  3. Our calculator offers complex differentiating into step-by-step solution.
  4. It gives a variety of differentiating calculus function solutions due to built-in differential.

Further, For an extensive collection of mathematical tools and calculators, including integral calculators, visit our all calculators section.

Take Care of These Errors While Using Derivatives Calculator

Users should check these errors which are mentioned below to get accurate or for better results.

  • Your expression of the function you would like to differentiate should not contain f(x)= part.
  • Make sure that the function depends on the differentiation variable you specified.
  • Ensure the expression you are differentiating is properly formatted and there is no red error above the input field.

Frequently Asked Question

How to Find the Derivative of a Function

Calculating the derivative of a function is also called as differentiation. In calculus, it helps to measure the change in functions as its input changes. There are many steps to calculate the derivative online but we will give some steps to find the derivative of the function manually,

  • The derivative shows the slope of the function or the rate of change in function on the given point.
  • To make the differentiation process smooth, simplify the function
  • Identify if the function is a trigonometric function, a simple polynomial, or an exponential function.
  • Apply any appropriate rule of differentiation on the function’s form
  • If the function is the difference of several terms or a sum then differentiate each term separately and then combine the results.
  • If there is any special function such as trigonometric, logarithmic, or exponential function then use their special differentiation rules.
  • Now apply the rules, combine the identical terms, and simplify the expression of the function to get the results.

How to Find the Derivative of a Graph

To find the derivative of a graph you should understand the relationship between the graph and its derivative. There are several ways to find derivative online for a graph but we will give some simple steps to calculate the derivative of the graph,

  • Draw a tangent line on the point where you want to determine the derivative. This line should touch the curve on the single point without going through it.
  • Determine the slope of the tangent line by choosing two points on the tangent line and the use the formula,

$$ \frac{y_2 - y_1}{x_2 - x_1} $$

  • Use the formal definition of derivative, if you have a function’s equation,

$$ f’(x) \;=\; lim{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} $$

  • Now interpret the derivative

What is the Derivative of Trigonometric Functions

The derivative of the trigonometric function is determining the rate of change of the trigonometric function concerning its variables. The derivatives of trigonometric functions are,

  • The derivative of sin(y) is cos(y)
  • The derivative ofcos(y) is -sin(y)
  • The derivative of tan(y) is sec2(y)
  • The derivative of cot(y) is -csc2(y)
  • The derivative of sec(y) is sec(y)tan(y)
  • The derivative of csc(y) is -csc(y)cot(y)

Such derivatives can be derived by using the limit definition of the derivative and different trigonometric identities.

How to Find the Derivative of an Integral

To find derivative with steps of the integral, which is also referred to as the second fundamental theorem of calculus, follow the given steps

  • For an indefinite integral, the derivative of the original function would be taken which can be expressed as,

$$ \frac{d}{dx} \int f(x) dx \;=\; f(x) $$

  • For a definite integral, if the integral has constant limits then its derivative would be zero as the integral determined to a constant value and the derivative of this constant is zero,

$$ \frac{d}{dx} \int_{a}^{b} f(t) dt \;=\; 0 $$

  • For a definite integral having a variable upper limit then the lower limit is a constant, the derivative of the integral with respect to x is an integrand determined at x,

$$ \frac{d}{dx} \int_{a}^{x} f(t) dt \;=\; f(x) $$

  • For a definite integral having variable limits, both the upper and the lower limits are functions of x, here you would apply the Leibniz rule which is,

$$ \frac{d}{dx} \int_{h(x)}^{g(x)} f(t) dt \;=\; f(g(x)) . g’(x) - f(h(x)) . h’(x) $$

What is the Derivative of Tan

The derivative of the function concerning x is,

$$ \frac{d}{dx} \;=\; sec^2(x) $$

This would be derived by using the quotient rule,

$$ \frac{d}{dx} (tan(x)) \;=\; \frac{d}{dx} \biggr( \frac{sin(x)}{cos(x)} \biggr) \;=\; \frac{cos(x) cos(x) + sin(x) sin(x)}{cos^2 (x)} \;=\; sec^2(x) $$

What is the Derivative of Tan^2(x)

The derivative of tan2 can be obtained by applying the chain rule of the function which is the composition of the functions and also be used to solve derivative online when some steps are followed. The steps to calculate the derivative of tan2 is,

  • Spot the outer function and the inner function
  • Now differentiate the outer function
  • Differentiate the inner function
  • Multiply the derivative of outer and inner functions by applying chain rule,

$$ \frac{d}{dx}(tan^2(x)) \;=\; 2u . \frac{du}{dx} \;=\; 2 tan(x) sec^2(x) $$

What is derivative of e ^2x?

The derivative of the function with respect to x is obtained by applying the chain rule of differentiation. The derivatives online of e 2x can solved using the following formula which is expressed as,

$$ \frac{d}{dx} e^{2x} \;=\; e^{2x} . \frac{d}{dx}(2x) \;=\; 2e^{2x} $$