Derivative Calculator

Power rule

Bring the exponent down as a factor, then reduce the exponent by one.

\[\frac{d}{dx}x^{n} = n x^{n-1}\]

What it says

For a whole number \(n\), the binomial theorem expands \((x + h)^{n}\) as \(x^{n} + n x^{n-1}h + \left(\text{terms with } h^{2} \text{ and higher}\right)\). Subtracting \(x^{n}\), dividing by \(h\) and letting \(h \to 0\) kills every term but \(n x^{n-1}\).

The rule is true for every real exponent, not only positive integers — negative, fractional and irrational exponents all obey it. That makes it the workhorse for roots and reciprocals, once they are rewritten as powers: \(\sqrt{x} = x^{1/2}\) and \(\frac{1}{x^{3}} = x^{-3}\).

The pattern is worth memorising in words rather than symbols: the old exponent becomes the coefficient, and the new exponent is one less.

When it applies

  • The base is the variable and the exponent is a constant.
  • Roots and reciprocals, after rewriting them with exponents.
  • Together with the chain rule when the base is a function: \(\left(u^{n}\right)' = n u^{n-1}u'\).

Five worked examples

Every line is the step the calculator would show, in the order it applies them. Each graph is live: hover it to read both curves and see the tangent whose slope is the derivative, drag to pan, scroll to zoom.

Example 1
\[\frac{d}{dx}\left[x^{5}\right]\]

  1. The exponent \(5\) comes down in front and drops to \(4\).

    \[\frac{d}{dx}\left[x^{5}\right] = 5x^{5-1}\]

  2. Tidy the exponent.

    \[= 5x^{4}\]
Answer
\[5 x^{4}\]

f is odd and very steep away from the origin. f′ is even and never dips below the axis, so f increases everywhere, touching zero only at \(x = 0\). That flat point is not a turning point — the curve carries on rising through it.

The plainest case: a positive whole-number exponent.

Open x^5 in the calculator
Example 2
\[\frac{d}{dx}\left[\sqrt{x}\right]\]

  1. Rewrite the root as a power.

    \[\sqrt{x} = x^{1/2}\]

  2. Apply the rule with \(n = \tfrac{1}{2}\).

    \[= \tfrac{1}{2}x^{-1/2}\]

  3. A negative exponent means a denominator.

    \[= \frac{1}{2\sqrt{x}}\]
Answer
\[\frac{1}{2 \sqrt{x}}\]

Both curves are cut off at the left, since neither exists for negative x. f′ plunges from a great height toward the axis, dropping below f near \(x = 0.5\).

Notice the derivative blows up as \(x \to 0^{+}\): the square-root graph is vertical at the origin.

Open sqrt(x) in the calculator
Example 3
\[\frac{d}{dx}\left[\frac{1}{x^{3}}\right]\]

  1. Move the power up top with a negative exponent.

    \[\frac{1}{x^{3}} = x^{-3}\]

  2. Bring \(-3\) down; \(-3 - 1 = -4\).

    \[= -3x^{-4}\]

  3. Back to fraction form.

    \[= -\frac{3}{x^{4}}\]
Answer
\[-\frac{3}{x^{4}}\]

A pole at \(x = 0\) splits the curve in two, with f′ negative on both branches: f falls wherever it is defined.

Reducing a negative exponent by one makes it more negative. This is the step most often got wrong.

Open 1/x^3 in the calculator
Example 4
\[\frac{d}{dx}\left[x^{\frac{2}{3}}\right]\]

  1. Apply the rule with \(n = \tfrac{2}{3}\).

    \[= \tfrac{2}{3}x^{2/3 - 1}\]

  2. Subtract the exponents.

    \[= \tfrac{2}{3}x^{-1/3}\]

  3. Rewrite as a cube root.

    \[= \frac{2}{3\sqrt[3]{x}}\]
Answer
\[\frac{2}{3 x^{\frac{1}{3}}}\]

Framed on the positive side, where both curves are real. f climbs with visibly decreasing steepness and f′ slides toward the axis.

Fractional exponents follow exactly the same mechanics — only the arithmetic is fiddlier.

Open x^(2/3) in the calculator
Example 5
\[\frac{d}{dx}\left[4 x^{7} - x^{2}\right]\]

  1. Sum rule splits the terms; constant multiple pulls out the \(4\).

    \[= 4\;\frac{d}{dx}\left[x^{7}\right] - \frac{d}{dx}\left[x^{2}\right]\]

  2. Power rule on each.

    \[= 4 \cdot 7x^{6} - 2x\]

  3. Multiply out.

    \[= 28x^{6} - 2x\]
Answer
\[28 x^{6} - 2 x\]

Near the origin the \(-x^{2}\) term wins and f dips below the axis; further out the seventh power takes over and it climbs steeply.

How the power rule usually shows up in practice: inside a sum, with coefficients along for the ride.

Open 4x^7 - x^2 in the calculator

Watch out for

  • If the variable is in the exponent — \(2^{x}\), \(x^{x}\) — this is the wrong rule entirely.
  • For a compound base such as \((3x + 1)^{4}\), the chain rule must supply the extra factor \(u'\).
  • \(x^{0} = 1\) is a constant, so its derivative is \(0\).
Chain ruleConstant multiple ruleExponential rule