Derivative Calculator

Sum rule

Differentiate term by term, then add the results back together.

\[\left(f + g\right)' = f' + g'\]

What it says

Differentiation distributes over addition: the derivative of a sum is the sum of the derivatives. It follows straight from the limit definition, because the limit of a sum is the sum of the limits.

Subtraction is covered as well, since \(f - g\) is \(f + (-1)g\) and the constant multiple rule handles the \(-1\). The rule extends to any number of terms with any signs.

In practice this is the first rule to apply to any long expression: split it into terms, handle each one with whatever rule fits it, then reassemble. It is what lets a fifteen-term polynomial be differentiated without a single hard step.

When it applies

  • A \(+\) or \(-\) joins terms at the top level of the expression.
  • Any polynomial, and any sum of unrelated functions such as \(e^{x} + \ln x\).
  • Before any other rule — split first, then choose a rule per term.

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
\[\frac{d}{dx}\left[x^{2} + x\right]\]

  1. Split into the two terms.

    \[\frac{d}{dx}\left[x^{2} + x\right] = \frac{d}{dx}\left[x^{2}\right] + \frac{d}{dx}\left[x\right]\]

  2. Power rule on the first, variable rule on the second.

    \[= 2x + 1\]
Answer
\[2 x + 1\]

f is a parabola and f′ the straight line \(2x + 1\). The line crosses zero at \(x = -\tfrac{1}{2}\), directly under the lowest point of the parabola. It is negative to the left of that and positive to the right: f falls, then rises.

Two different rules, one for each term — that is the whole point of splitting first.

Open x^2 + x in the calculator
Example 2
\[\frac{d}{dx}\left[x^{3} - 4 x^{2} + 7 x - 9\right]\]

  1. Four terms, four derivatives.

    \[= \frac{d}{dx}\left[x^{3}\right] - \frac{d}{dx}\left[4x^{2}\right] + \frac{d}{dx}\left[7x\right] - \frac{d}{dx}\left[9\right]\]

  2. Power rule with the coefficients carried along, and the constant term drops out.

    \[= 3x^{2} - 8x + 7 - 0\]
Answer
\[3 x^{2} - 8 x + 7\]

A cubic with a parabola for its derivative. The parabola never touches the axis, so f′ stays positive and f climbs without ever turning.

Every polynomial derivative is this: sum rule on the outside, power rule and constant multiple inside.

Open x^3 - 4x^2 + 7x - 9 in the calculator
Example 3
\[\frac{d}{dx}\left[\sin\left(x\right) + \cos\left(x\right)\right]\]

  1. Split the sum.

    \[= \frac{d}{dx}\left[\sin x\right] + \frac{d}{dx}\left[\cos x\right]\]

  2. The two trigonometric derivatives, keeping the minus sign that \(\cos\) produces.

    \[= \cos x + \left(-\sin x\right)\]
Answer
\[\cos\left(x\right) - \sin\left(x\right)\]

Two waves of equal amplitude, a quarter-turn apart: f′ peaks where f crosses its midline and reads zero at each peak of f.

The minus sign comes from the derivative of \(\cos\), not from the sum rule.

Open sin(x) + cos(x) in the calculator
Example 4
\[\frac{d}{dx}\left[e^{x} + \ln\left(x\right)\right]\]

  1. Split.

    \[= \frac{d}{dx}\left[e^{x}\right] + \frac{d}{dx}\left[\ln x\right]\]

  2. Exponential rule and logarithm rule.

    \[= e^{x} + \frac{1}{x}\]
Answer
\[e^{x} + \frac{1}{x}\]

Near the origin the logarithm dominates and f′ is enormous; further right the exponential takes over and both curves sweep upwards together.

The terms need not be related in any way — each is differentiated in isolation.

Open e^x + ln(x) in the calculator
Example 5
\[\frac{d}{dx}\left[\sqrt{x} + \frac{1}{x}\right]\]

  1. Rewrite both terms as powers so the power rule applies.

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

  2. Differentiate each.

    \[= \tfrac{1}{2}x^{-1/2} + \left(-1\right)x^{-2}\]

  3. Back to root and fraction notation.

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

Only for \(x > 0\). f falls steeply, bottoms out near \(x = 1.59\) where f′ crosses zero, then rises slowly.

Rewriting before splitting often saves work; here it turns two awkward terms into two power-rule applications.

Open sqrt(x) + 1/x in the calculator

Watch out for

  • There is no product analogue: \(\left(fg\right)' \neq f'g'\). Use the product rule.
  • A constant term contributes \(0\), not the constant itself.
  • \(\frac{f + g}{h}\) does split into \(\frac{f}{h} + \frac{g}{h}\), but \(\frac{f}{g + h}\) does not split at all.
Constant multiple rulePower ruleProduct rule