Derivative Calculator
The tool below finds the derivative of an expression and shows each step. Enter a function and the calculator will identify the appropriate rules — power rule, product rule, chain rule, and so on — and apply them one at a time.
Type naturally or click ⌨ in the box to use the math keyboard.
Examples: x^2 for x², \sin(x) for sine, e^x for the exponential.
The derivative of a function measures its instantaneous rate of change. For \(f(x) = x^2\), the derivative is \(f'(x) = 2x\) — the slope of the tangent line at any value of \(x\). Different functions need different rules: power rule, product rule, quotient rule, and chain rule.
Worked Examples
Three examples worked by hand — the same logic the calculator uses, just spelled out so you can follow it on paper.
Example 1: Power rule — differentiate \(f(x) = x^3 + 4x\)
Apply the power rule \(\frac{d}{dx}(x^n) = nx^{n-1}\) to each term, and use the fact that the derivative of a sum is the sum of derivatives:
\[f'(x) = 3x^2 + 4\]
The \(4x\) becomes \(4 \cdot 1 \cdot x^0 = 4\). Constants out front stay; the exponent drops by one.
Example 2: Product rule — differentiate \(f(x) = x^2 \sin(x)\)
This is a product of two functions, \(u = x^2\) and \(v = \sin(x)\). The product rule says \((uv)' = u'v + uv'\):
\[u' = 2x, \quad v' = \cos(x)\] \[f'(x) = 2x \sin(x) + x^2 \cos(x)\]
Example 3: Chain rule — differentiate \(f(x) = \sin(x^2)\)
A function inside another function uses the chain rule: differentiate the outer function, leave the inner function alone, then multiply by the derivative of the inner function.
Outer is \(\sin(,\cdot,)\), inner is \(x^2\). So:
\[f'(x) = \cos(x^2) \cdot \frac{d}{dx}(x^2) = \cos(x^2) \cdot 2x = 2x \cos(x^2)\]
What is a Derivative?
The derivative of a function \(f(x)\) measures its instantaneous rate of change — how fast the output changes when you wiggle the input. Geometrically, the derivative at a point is the slope of the tangent line to the graph at that point.
Different functions need different rules to differentiate. A polynomial uses the power rule. A product of two functions uses the product rule. A function inside a function uses the chain rule. The calculator above identifies which rules apply and shows the work step by step.
Try These Examples
x^2 + 3x— power rule\sin(x)— common trig derivativee^{2x}— chain rule with exponentialx^3 \cdot \cos(x)— product rule(x^2 + 1)/(x - 1)— quotient rule\sqrt{x^2 + 1}— chain rule with a root
Common Derivative Rules
A quick reference for the rules this calculator uses:
| Function | Derivative |
|---|---|
| \(x^n\) | \(nx^{n-1}\) |
| \(e^x\) | \(e^x\) |
| \(a^x\) | \(a^x \ln a\) |
| \(\ln(x)\) | \(\dfrac{1}{x}\) |
| \(\sin(x)\) | \(\cos(x)\) |
| \(\cos(x)\) | \(-\sin(x)\) |
| \(\tan(x)\) | \(\sec^2(x)\) |
| \(k\) (constant) | \(0\) |
And the three combining rules for any two functions \(u\) and \(v\):
- Product rule: \((uv)' = u'v + uv'\)
- Quotient rule: \(\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}\)
- Chain rule: \(\bigl(f(g(x))\bigr)' = f'(g(x)) \cdot g'(x)\)
Tips for Using the Calculator
- For powers, use
^:x^3means \(x^3\) - For fractions, use
\frac{a}{b}or justa/b - For square roots, use
\sqrt{x} - Trig functions:
\sin(x),\cos(x),\tan(x) - The natural exponential is
e^x; the natural log is\ln(x) - The virtual math keyboard gives one-click access to all common symbols
If you want to reverse the process — find an antiderivative or integral — try the Antiderivative Calculator or Integral Calculator.
Frequently Asked Questions
What does the derivative represent geometrically?
The derivative at a point \(x = a\) is the slope of the tangent line to the graph at that point. If the derivative is positive, the function is increasing there; if negative, decreasing; if zero, the graph has a horizontal tangent (often a local max, min, or inflection point).
Why is the derivative of a constant zero?
A constant function doesn't change as \(x\) changes — its graph is a horizontal line. The slope of a horizontal line is zero, so the rate of change is zero. Algebraically, \(\frac{d}{dx}(k) = 0\) for any constant \(k\).
When do I use the chain rule vs the product rule?
Use the product rule when two functions are multiplied: \(x^2 \cdot \sin(x)\) is a product. Use the chain rule when one function is inside another: \(\sin(x^2)\) is a composition — sine of \(x^2\). Look at whether the functions are side by side (product) or nested (chain).
What's a higher-order derivative?
A higher-order derivative is the derivative of a derivative. The second derivative \(f''(x)\) is the derivative of \(f'(x)\); the third \(f'''(x)\) is the derivative of \(f''(x)\); and so on. The second derivative measures the concavity of a function and is what you use to test whether a critical point is a local max or min.
What's the difference between a derivative and an integral?
Differentiation finds the rate of change of a function. Integration is the reverse: given a rate of change, integration finds the original function (an antiderivative). They're inverse operations, formalized by the Fundamental Theorem of Calculus.