Derivative Calculator

This free derivative calculator finds the derivative of any function and shows each step. It names the rule it uses, such as the power rule, product rule, quotient rule, or chain rule. You can also find second, third, or higher derivatives, and get the slope at a point.

How to Use the Derivative Calculator

1. Type your function, for example x^2 sin(x).
2. Choose the variable and the order (1 for the first derivative, 2 for the second, and so on).
3. Optional: type a number in Evaluate at x = to get the slope at that point.
4. Press Solve. Open the steps to see every rule, and the graph to compare \(f\) with \(f’\).

What Is a Derivative?

A derivative tells you how fast a function is changing. On a graph, it is the slope of the tangent line at each point. If \(f(x)\) is the position of a car, then \(f'(x)\) is its speed.

The formal definition uses a limit:

\[f'(x) = \lim_{h\to 0} \frac{f(x+h) – f(x)}{h}.\]

In practice, we use rules instead of this limit. The rules are faster and give the same answer. Derivatives and antiderivatives undo each other. See antiderivative vs derivative for a side-by-side look.

Derivative Rules

Rule	Formula	Example
Constant	\(\frac{d}{dx}c = 0\)	\(\frac{d}{dx}7 = 0\)
Power	\(\frac{d}{dx}x^n = nx^{n-1}\)	\(\frac{d}{dx}x^5 = 5x^4\)
Constant multiple	\((cf)’ = cf’\)	\(\frac{d}{dx}4x^3 = 12x^2\)
Sum	\((f+g)’ = f’ + g’\)	\(\frac{d}{dx}(x^2 + x) = 2x + 1\)
Product	\((fg)’ = f’g + fg’\)	\(\frac{d}{dx}x\sin x = \sin x + x\cos x\)
Quotient	\(\left(\frac fg\right)’ = \frac{f’g – fg’}{g^2}\)	\(\frac{d}{dx}\frac{x}{x+1} = \frac{1}{(x+1)^2}\)
Chain	\(\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)\)	\(\frac{d}{dx}\sin(3x) = 3\cos(3x)\)

Derivatives of Common Functions

Function	Derivative	Function	Derivative
\(\sin x\)	\(\cos x\)	\(e^x\)	\(e^x\)
\(\cos x\)	\(-\sin x\)	\(a^x\)	\(a^x \ln a\)
\(\tan x\)	\(\sec^2 x\)	\(\ln x\)	\(\frac{1}{x}\)
\(\sec x\)	\(\sec x\tan x\)	\(\sqrt{x}\)	\(\frac{1}{2\sqrt{x}}\)
\(\tan^{-1} x\)	\(\frac{1}{1+x^2}\)	\(\sin^{-1} x\)	\(\frac{1}{\sqrt{1-x^2}}\)

Worked Examples

Example 1: Product rule, \(\frac{d}{dx}\left[x^2\sin x\right]\)

Let \(f = x^2\) and \(g = \sin x\). Then \(f’ = 2x\) and \(g’ = \cos x\).

\((fg)’ = 2x\sin x + x^2\cos x\).

Example 2: Chain rule, \(\frac{d}{dx}\,e^{3x^2}\)

The outside is \(e^u\) and the inside is \(u = 3x^2\). The derivative of the outside is \(e^u\). The derivative of the inside is \(6x\).

Answer: \(6x\,e^{3x^2}\).

Example 3: Quotient rule, \(\frac{d}{dx}\frac{\sin x}{x}\)

\(\frac{(\cos x)(x) – (\sin x)(1)}{x^2} = \frac{x\cos x – \sin x}{x^2}\).

Example 4: Logarithmic differentiation, \(\frac{d}{dx}\,x^x\)

Here both the base and the exponent change, so no basic rule fits. Take the log: \(\ln y = x\ln x\). Differentiate both sides: \(\frac{y’}{y} = \ln x + 1\). So \(y’ = x^x(\ln x + 1)\).

Common Mistakes

• Forgetting the chain rule. \(\frac{d}{dx}\sin(2x)\) is \(2\cos(2x)\), not \(\cos(2x)\).
• Multiplying derivatives in a product. \((fg)’ \ne f’g’\). Use \(f’g + fg’\).
• Mixing up the quotient rule order. It is “low d-high minus high d-low, over low squared.”
• Treating \(e^x\) like \(x^n\). \(\frac{d}{dx}e^x = e^x\), not \(xe^{x-1}\).

Learn the Math Behind the Derivative Calculator

A derivative measures how fast a function changes. Every antiderivative can be checked by taking a derivative.

• GuideAntiderivative vs derivative, side by side
• GuideHow to check an integral by differentiating
• GuideHow to study calculus: a roadmap with checkpoints
• Key termsDerivative, Chain rule, Product rule, Quotient rule
• Next toolsSecond Derivative Calculator, Partial Derivative Calculator, Instantaneous Rate of Change Calculator, Antiderivative Calculator

Frequently Asked Questions

Related Guides

• Antiderivative vs derivative: what is the difference?
• How to study calculus on your own
• The power rule (in reverse) for antiderivatives

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