Derivative Calculator: Calculate Derivatives

This Derivative Calculator is a free online tool that can help find the derivative of various functions. It makes finding derivatives quick and easy.

A derivative is used in mathematics to find out how fast something is changing. A derivative is one of the most important concepts in calculus. Derivatives can be found throughout mathematics, physics, engineering, economics, statistics and many more scientific fields.

For example, if we have a function describing how far a vehicle has travelled over time, the derivative will show us the instantaneous velocity of that vehicle. If we have a function describing the position of an object, we can find the velocity and acceleration of it using the derivative.

This Derivative Calculator makes it easier to find derivatives. We input our function and it finds the derivative for us.

This can be useful when trying to find derivatives of polynomials, exponentials, logarithmic functions and trigonometric functions.

This can be very helpful for students in the United States whilst studying calculus in highschool, AP Calculus, maths in college and engineering.

This can make doing your calculus homework, revision and checking your maths much easier.

How to Use a Derivative Calculator

This Derivative Calculator is easy to use.

To find the derivative of your function follow the steps below.

1. Input Function

Type in your mathematical function. For example:
x^2 + 3x + 5

2. Pick your variable

Pick the variable you want to differentiate with respect to. Usually this will be x.

3. Choose order of derivative

If the calculator allows you to choose, select whether you want to find the first, second or higher order derivative.

4. Click ‘Calculate’

Click calculate or solve.

5. See your answer

The calculator will show you the derivative of your function.

6. See your solution

If there are step by step calculations then you can see how your function was differentiated.

7. Do another calculation

Clear the previous function and type in a new function.

Example

Type in:
f(x) = x^2 + 3x + 5
Once you have calculated the derivative you will get:
f’(x) = 2x + 3
This shows us how quickly the original function is changing at any point in x.

Common Rules of Differentiation

Different rules of differentiation are used for different types of functions.

1. Power Rule

This is commonly used when differentiating polynomial functions.
d/dx(x^n) = nx^(n-1)
Example:
d/dx(x^4) = 4x^3

2. Constant Rule

The derivative of a constant will always be 0.
d/dx(c) = 0
Example:
d/dx(10) = 0

3. Sum/Difference Rule

The derivative of a sum or difference will be the sum or difference of the individual derivatives.
d/dx(f(x) ± g(x)) = f’(x) ± g’(x)

4. Product Rule

When differentiating a product of functions use:

\\((fg)\' = f\'g + fg\'\\)
\\(\\Delta x\\)
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f × g f × Δg g × Δf Δf × Δg f(x) Δf g(x) Δg
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5. Quotient Rule

When differentiating a quotient use:
d/dx(f/g) = (gf’ - fg’) / g^2

6. Chain Rule

When differentiating a composite function use:
d/dx(f(g(x))) = f’(g(x))g’(x)
These rules can be used to easily solve different types of differentiation problems.

Example Calculation

Let’s take a look at how a Derivative Calculator works on a simple polynomial function.

Find the derivative of:

f(x) = 3x^3 + 4x^2 + 5x + 7
Step 1: Identify the function.
We have 3 terms with variables and a constant.
f(x) = 3x^3 + 4x^2 + 5x + 7
Step 2: Use the power rule.
Now differentiate each term.
First term:
d/dx(3x^3) = 9x^2
Second term:
d/dx(4x^2) = 8x
Third term:
d/dx(5x) = 5
Fourth term:
d/dx(7) = 0
Step 3: Put it all together.
f’(x) = 9x^2 + 8x + 5 + 0
Step 4: Give your answer.
Final Answer
f’(x) = 9x^2 + 8x + 5

This now shows us how quickly the original function is changing at any point in time.

For example, when x = 2:
f’(2) = 9(2)^2 + 8(2) + 5
= 36 + 16 + 5
f’(2) = 57
This tells us that the instantaneous rate of change at \\(\\frac{f(1 + 2) - f(1)}{2} = 4\\quad \\text{error} = 2\\)0 is 57.

Understanding your result

The result given will be the derivative of the original function.

Depending on what you have calculated you may get different results.

• First Derivative: Shows the instantaneous rate of change or how steep the function is.
• Second Derivative: Shows how the first derivative is changing. This can be useful for identifying concave/convex and acceleration.
• Higher Order Derivatives: Shows how many times the function has been differentiated.
• Simplified: Shows the derivative in its simplest form.
• Constant: Shows that the derivative is a fixed number.
• Zero: Shows that there is no instantaneous change in the function.

For example:
f(x) = x^3
First Derivative:
f’(x) = 3x^2
Second Derivative:
f\'\'(x) = 6x

The first derivative shows how steep the function is and the second derivative shows how this is changing.

Things that could effect your result

There are a few things that could affect your result.

• Original Function: The function you type in will determine your result. If you put in any incorrect numbers, variables or symbols you may get unexpected results.
• Variable to differentiate: If your function has more than one variable you may get different results depending on what variable you choose.
• Order of derivative: Depending on whether you choose first, second or higher order derivatives your result will be different.
• Type of function: Depending on what type of function you have, polynomials, exponentials, logarithmic, trigonometric or rational you will use different rules.
• Symbols: You must use the correct symbols for your function. Brackets, exponents, multiplication and division.
• Chain Rule: If your function is a composite function you must use the chain rule. If you don’t use this correctly you may get an inaccurate result.
• Domain: Some functions cannot be differentiated at all points. Division by zero, discontinuity and sharp points can affect the derivative.
• Numerical: Some calculators use numerical methods so your result may be slightly different.

Advantages and Uses of a Derivative Calculator

This Derivative Calculator can be very advantageous.

• Saves Time: This Derivative Calculator can quickly calculate derivatives.
• Helpful for students: This can be helpful for students studying calculus, algebra and higher level maths to check their homework.
• Useful for AP Calculus: Students taking the AP Calculus AB and BC exams can use this to revise differentiation rules and check practice questions.
• Understand functions: Differentiation can help to find out where functions are increasing/decreasing, maximums, minimums and turning points.
• Useful in Physics: Derivatives can be used to calculate velocity, acceleration and rate of change.
• Helpful for engineers: Engineers use derivatives when analysing movement, optimisation, changing quantities and modelling.
• Useful in Economics: Businesses and economists use derivatives to understand marginal cost, marginal revenue, optimising profits and changing variables.
• Help with Optimisation: Derivatives can help find critical points which could be maximum/minimum.

Most Derivative Calculators in browsers can be used on computers, laptops, tablets and smartphones.

Accuracy, Assumptions and Limitations

If you type in the correct function and the calculator can calculate the derivative you require then you will get an accurate result.

However there are some assumptions and limitations.

• Correct Input: If you type in incorrect maths you will get incorrect/unexpected results.
• Not all functions: Some calculators may not have advanced functions, implicit differentiation or complex composite functions.
• Not differentiable: There may be points where you cannot differentiate.
• Domain: Logarithmic, fractional and square root functions may have restrictions on their domain.
• Simplify: Different expressions may simplify differently.
• Numerical: Numerical methods can cause rounding errors.
• Learning: Whilst this can give you the answers, it is important to understand the rules of differentiation to learn calculus.

If doing important work for your studies/profession then check the result using differentiation rules and if the function can be differentiated at the point required.

FAQs

1. What is a Derivative Calculator?

A Derivative Calculator is an online tool that can calculate the derivative of a function. This can help solve different problems involving differentiation of polynomials, exponentials, logarithmic functions and trigonometric functions depending on the calculator.

2. Is it free?

Yes! Many Derivative Calculators online are free. You can type in your maths functions and get the derivatives.

3. Can it calculate second derivatives?

Yes! If the calculator has higher order differentiation you can calculate second, third and other derivatives. The second derivative is commonly used to find concave, acceleration and how the gradient is changing.

4. What is the difference between a derivative and an integral?

A derivative is used to find the instantaneous rate of change, whereas integrals are used to find accumulations such as area under curves.

5. Can I use it on my smartphone?

Yes! Most Derivative Calculators in browsers can be used on smartphones, tablets, laptops and desktops. You can simply open your browser.