Derivative Calculator

Category: Calculus

- April 01, 2025

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Calculate the derivative of mathematical functions. This calculator supports common functions including polynomials, trigonometric, logarithmic, and exponential expressions.

Function Input

Function f(x):

Order of Derivative:

Variable:

Evaluation (Optional)

Evaluate derivative at a specific point

Display Options

Output Format:

Show differentiation rules

Show graph

About Derivatives

In calculus, the derivative measures how a function changes as its input changes. Specifically, the derivative of a function at a point measures the rate at which the function's output changes per unit change in its input.

Common Differentiation Rules

Power Rule: If f(x) = xⁿ, then f'(x) = n·x^n-1
Sum Rule: If h(x) = f(x) + g(x), then h'(x) = f'(x) + g'(x)
Product Rule: If h(x) = f(x)·g(x), then h'(x) = f'(x)·g(x) + f(x)·g'(x)
Quotient Rule: If h(x) = f(x)/g(x), then h'(x) = [f'(x)·g(x) - f(x)·g'(x)]/[g(x)]²
Chain Rule: If h(x) = f(g(x)), then h'(x) = f'(g(x))·g'(x)

Trigonometric Functions

d/dx[sin(x)] = cos(x)
d/dx[cos(x)] = -sin(x)
d/dx[tan(x)] = sec²(x)
d/dx[sec(x)] = sec(x)·tan(x)
d/dx[csc(x)] = -csc(x)·cot(x)
d/dx[cot(x)] = -csc²(x)

Exponential and Logarithmic Functions

d/dx[e^x] = e^x
d/dx[a^x] = a^x·ln(a)
d/dx[ln(x)] = 1/x
d/dx[log_a(x)] = 1/(x·ln(a))

Basic Derivative Formula:
\[ \frac{d}{dx}f(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]

What Is the Derivative Calculator?

The Derivative Calculator is an easy-to-use online tool that helps you find derivatives of mathematical functions instantly. Whether you're a student studying calculus or a professional needing quick differentiation results, this tool saves time and effort by providing accurate answers, step-by-step solutions, and visual graphs.

It supports common function types like polynomials, trigonometric, exponential, and logarithmic expressions. You can also explore higher-order derivatives, compute specific values at points, and switch between simplified, detailed, or LaTeX formats.

How to Use the Derivative Calculator

Follow these simple steps to compute a derivative:

Enter a function, such as x^2 + sin(x), in the input box labeled f(x).
Select the order of the derivative you want to calculate (first, second, third, etc.).
Choose the variable you want to differentiate with respect to (x, y, t, etc.).
(Optional) Check the box to evaluate the derivative at a specific point and enter the value.
Pick how you’d like the result to appear: simplified, with steps, or as a LaTeX formula.
Click “Calculate Derivative” to see results and graph.

Key Features

Instant derivative results with real-time calculation.
Second derivative solver and up to fifth-order derivatives supported.
Graphical representation of both the original function and its derivative.
Detailed steps showing how the derivative was computed.
Includes LaTeX output for math-friendly formats.
Shows applied differentiation rules like the Chain Rule, Product Rule, and more.

Who Can Benefit From This Calculator?

This differentiation tool is especially useful for:

Students learning calculus or needing help with homework.
Teachers and tutors who want to explain concepts with clear steps and graphs.
Engineers and scientists performing quick mathematical analysis.
Anyone needing to solve derivatives online quickly and accurately.

More Than Just First Derivatives

This calculator is more than a basic tool—it can also assist with more advanced topics:

Second Derivative Calculator: Analyze acceleration or concavity.
nth Derivative Calculator: Compute higher-order derivatives for deeper analysis.
Partial Derivative Calculator: Solve for multivariable functions (e.g., ∂f/∂x).
Implicit Derivative Calculator: Differentiate equations where y is not isolated.
Directional Derivative Calculator: Examine the rate of change in any given direction.

FAQs

Can I evaluate the derivative at a specific point?

Yes, just check the option labeled "Evaluate derivative at a specific point" and enter the desired x-value.

What types of functions does it support?

It supports polynomials, trig functions like sin(x), exponential functions like e^x, and logarithmic functions like ln(x).

Can I compute higher-order derivatives?

Yes, you can find up to the fifth derivative using the dropdown menu for derivative order.

Does it show steps and rules used?

Yes, if you enable the options, the calculator will show differentiation steps and which rules were applied.

Can it graph the function and its derivative?

Absolutely. It provides an interactive graph showing both curves for comparison and better understanding.

Does it work for multivariable functions?

Yes, use it as a partial derivative solver by selecting the variable you want to differentiate with respect to.

Why Use This Calculator?

Whether you're trying to solve derivatives online, find second derivatives for curve analysis, or need a fast derivative solver for class or work, this tool offers speed, accuracy, and clarity. It's also great for visual learners thanks to its built-in graphing feature.

Looking for more? Try out related tools like the Integral Calculator to find antiderivatives, the Tangent Line Calculator for slope analysis, or the Limit Calculator to evaluate tricky limit expressions.