Derivative Calculator

A derivative measures how fast a function changes: the slope of its graph at each point. This derivative calculator differentiates functions of x using the power, product, quotient and chain rules, showing which rule is used at each step. It can also find the slope and tangent line at a chosen point.

Use x as the variable. Functions: sin, cos, tan, ln, log, sqrt, exp, e^x, abs.
Try:

How to use the derivative calculator

  1. Type the function using x, such as x^3 - 4x + 1 or sin(2x).
  2. Optionally, enter an x-value to get the slope and tangent line there.
  3. Press Calculate to see the derivative, every rule used, and the graph.

Formula

ddxxn=nxn−1\frac{d}{dx}x^n = nx^{n-1}
ddxf(g(x))=f′(g(x)) g′(x)\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)

Worked example: x³ − 4x + 1

  • f(x) =: x^3 - 4x + 1
  • Evaluate at x = (optional): 2

✓ Answer checked

ddx[x3−4x+1]=3x2−4\frac{d}{dx}\left[x^{3} - 4x + 1\right] = 3x^{2} - 4
f'(2)
88
Tangent line
y=8(x−2)+1y = 8(x - 2) + 1
−2−112345650100150slope 8f(x)
The tangent to f at x = 2 has slope 8, the value of the derivative there.
Step-by-step working (6 steps)

  1. Read the function

    This is how the input was understood.

    f(x)=x3−4x+1f(x) = x^{3} - 4x + 1

  2. Sum rule Sum/difference rule

    Differentiate each term separately.

    ddx[x3−4x+1]=3x2−4\frac{d}{dx}\left[x^{3} - 4x + 1\right] = 3x^{2} - 4

  3. Sum rule Sum/difference rule

    Differentiate each term separately.

    ddx[x3−4x]=3x2−4\frac{d}{dx}\left[x^{3} - 4x\right] = 3x^{2} - 4

  4. Constant multiple Constant multiple rule

    A constant factor stays in front.

    ddx[4x]=4\frac{d}{dx}\left[4x\right] = 4

  5. Power rule (xⁿ)' = nxⁿ⁻¹

    Bring the power down and reduce it by one.

    ddx[x3]=3x2\frac{d}{dx}\left[x^{3}\right] = 3x^{2}

  6. Simplify

    Tidy up the result.

    f′(x)=3x2−4f'(x) = 3x^{2} - 4
Formulas used
ddxxn=nxn−1\frac{d}{dx}x^n = nx^{n-1}
(uv)′=u′v+uv′(uv)' = u'v + uv'
(uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}
ddxf(g(x))=f′(g(x)) g′(x)\frac{d}{dx}f(g(x)) = f'(g(x))\,g'(x)
How this was checked
  • ✓ Compared with a numerical derivative at 6 points (0.37, 1.13, 2.71, -0.83, 1.9, 0.61); they agree.

Frequently asked questions

What is a derivative?

The derivative of f(x) gives the slope of the graph of f at every point. It tells you the instantaneous rate of change.

When do I use the chain rule?

When one function is inside another, such as sin(2x) or (x² + 1)⁵. Differentiate the outside, keep the inside, then multiply by the derivative of the inside.

How is the answer checked?

The derivative is compared with a numerical estimate of the slope at several points. If they disagree, the answer is not shown.

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