# Derivative Calculator

## Calculate derivatives step by step

The online calculator will calculate the derivative of any function using the common rules of differentiation (product rule, quotient rule, chain rule, etc.), with steps shown. It can handle polynomial, rational, irrational, exponential, logarithmic, trigonometric, inverse trigonometric, hyperbolic, and inverse hyperbolic functions. Also, it will evaluate the derivative at the given point if needed. It also supports computing the first, second, and third derivatives, up to 10.

Related calculators: Logarithmic Differentiation Calculator, Implicit Differentiation Calculator with Steps

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The Derivative Calculator is an online tool designed to calculate the derivative of a given function. In mathematics, the derivative shows how the value of a function changes when its input changes.

## How to Use the Derivative Calculator?

• ### Input

Enter the mathematical function for which you want to calculate the derivative. Ensure that your function is correctly formatted. Specify the variable with respect to which you want to calculate the derivative. (This step is optional.)

• ### Calculation

Once you've inputted your function, click the "Calculate" button.

• ### Result

You'll get the derivative of your function with respect to the chosen variable.

## What Is a Derivative?

The derivative is a fundamental concept in calculus that provides a precise way to understand and quantify how the values of functions change when their input values change.

The derivative of a function $f(x)$ with respect to the variable $x$ can be denoted differently:

• Leibniz Notation: $\frac{dy}{dx}$ or $\frac{d}{dx}\left(f(x)\right)$.
• Lagrange (Prime) Notation: $y^{\prime}$ or $f^{\prime}(x)$.
• Euler Notation: $\left(Df\right)(x)$.

The derivative of a function $f(x)$ with respect to $x$ represents the rate of change of $f$ at the given point $x$. It is defined as

$$f^{\prime}(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$$

For example, let $f(x)=x^2$. We want to find $f^{\prime}(x)$.

1. Use the derivative formula:

$$f^{\prime}(x)=\lim_{h\to0}\frac{(x+h)^2-x^2}{h}$$
2. Expand the numerator:

$$f^{\prime}(x)=\lim_{h\to0}\frac{x^2+2xh+h^2-x^2}{h}$$
3. Now, cancel out the $x^2$ terms:

$$f^{\prime}(x)=\lim_{h\to0}\frac{2xh+h^2}{h}$$
4. Factor out $h$ from the numerator:

$$f^{\prime}(x)=\lim_{h\to0}\frac{h\left(2x+h\right)}{h}$$
5. Cancel out $h$ in the numerator and denominator:

$$f^{\prime}(x)=\lim_{h\to0}\left(2x+h\right)$$
6. Now, let $h$ approach zero:

$$f^{\prime}(x)=2x+0=2x$$

So the derivative of $f(x)=x^2$ is $f^{\prime}(x)=2x$.

Derivative Rules:

• Constant Rule: The derivative of a constant is zero.

Example: $f(x)=5$ implies $f^{\prime}(x)=0$.

• Power Rule: The derivative of a power function is $\left(x^n\right)^{\prime}=nx^{n-1}. Example:$\left(x^3\right)^{\prime}=3x^{3-1}=3x^2$. • Constant Multiple Rule: The derivative of a function times a constant equals the constant times the derivative of the function:$\left(cf(x)\right)^{\prime}=c\left(f(x)\right)^{\prime}$. Example:$\left(5x^2\right)^{\prime}=5\left(x^2\right)^{\prime}=5\cdot2x=10x$. • Sum Rule: The derivative of the sum of two functions is the sum of their derivatives, i.e.$\left(f(x)+g(x)\right)^{\prime}=f^{\prime}(x)+g^{\prime}(x)$. Example:$\left(x^2+7x\right)^{\prime}=\left(x^2\right)^{\prime}+\left(7x\right)^{\prime}=2x+7$. • Product Rule: The derivative of the product of two functions is more complex:$\left(f(x)g(x)\right)^{\prime}=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)$. Example:$\left(x\sin(x)\right)^{\prime}=\left(x\right)^{\prime}\sin(x)+x\left(\sin(x)\right)^{\prime}=\sin(x)+x\cos(x)$. • Quotient Rule: The derivative of the quotient is given by the following formula:$\left(\frac{f(x)}{g(x)}\right)^{\prime}=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{g^2(x)}$. Example:$\left(\frac{x}{\sin(x)}\right)^{\prime}=\frac{\left(x\right)^{\prime}\sin(x)-x\left(\sin(x)\right)^{\prime}}{\sin^2(x)}=\frac{\sin(x)-x\cos(x)}{\sin^2(x)}$. • Chain Rule: For composite functions, the chain rule applies. If$u(x)=g\left(f(x)\right)$, then$u^{\prime}(x)=g^{\prime}\left(f(x)\right)f^{\prime}(x)$. Example:$\left(\cos\left(x^2\right)\right)^{\prime}=-\sin\left(x^2\right)\left(x^2\right)^{\prime}=-\sin\left(x^2\right)\cdot2x=-2x\sin\left(x^2\right)\$.

The derivative is a powerful tool for analyzing changes in functions and has wide applications in mathematics and science.

## Why Choose Our Derivative Calculator?

• ### Accuracy

Our calculator ensures accurate derivative calculations.

• ### User-Friendly Interface

Our user-friendly interface is designed to make the process as simple as possible. You don't need to be a math expert to use our tool effectively.

• ### Support for Various Functions

Our derivative calculator supports a wide range of functions, from basic polynomials to complex trigonometric, exponential and logarithmic functions. It is a universal tool for various mathematical problems.

• ### Step-by-Step Solutions

If you're studying calculus or want to enhance your math skills, our calculator offers a valuable learning experience.

### FAQ

#### What is a derivative in math?

In mathematics, a derivative represents the rate at which a function changes at a specific point. It provides the slope of the tangent line to the curve of a function at that point. It measures how a function's output changes when its input changes, offering valuable insight into the function's behavior.

#### Why is the derivative important?

Derivatives are critical in mathematics and various fields such as physics, engineering, economics, and biology. They provide insight into how quantities change over time relative to other variables.

#### What types of functions does the calculator support?

Our calculator supports a wide range of functions, from basic polynomials to complex trigonometric, exponential, and logarithmic expressions. You can use it for various mathematical functions found in calculus.

#### What if I encounter issues or have questions while using the calculator?

If you encounter any issues or have questions, feel free to reach out to our support team for assistance. We're here to help you get the most out of your Derivative Calculator.