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Derivative calculator

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  • Arithmetic to calculus

A derivative is a slope: how fast the output moves when the input nudges. Four rules cover almost everything you will be asked to differentiate, and the skill is spotting which shape you are looking at before you start.

The four rules, and how to spot them

Read the function from the outside in. The outermost operation decides the first rule you apply.

  1. Power rule

    Bring the exponent down, subtract one from it. The derivative of xⁿ is n·xⁿ⁻¹, and it works for negative and fractional powers too.

  2. Product rule

    Two things multiplied: first times the derivative of the second, plus second times the derivative of the first.

  3. Quotient rule

    A fraction: (bottom × derivative of top − top × derivative of bottom) ÷ bottom squared. The order of that subtraction matters.

  4. Chain rule

    A function inside another: differentiate the outside, leave the inside alone, then multiply by the derivative of the inside. This is the one that gets forgotten.

  5. Implicit differentiation

    When y is tangled up with x, differentiate both sides with respect to x, remembering that every y term picks up a dy/dx, then solve for dy/dx.

Worked examples

Each one is solved the way the solver solves it: the rule first, then the line.

Find the derivative of x^3 * sin(x)

ddx(x3sin⁡x)\frac{d}{dx}\big( x^3 \sin x \big)
  1. Two factors, so use the product rule
    (uv)′=u′v+uv′(uv)' = u'v + uv'
  2. Differentiate each part
    u=x3⇒u′=3x2,v=sin⁡x⇒v′=cos⁡xu = x^3 \Rightarrow u' = 3x^2, \qquad v = \sin x \Rightarrow v' = \cos x
  3. Put it together
    3x2sin⁡x+x3cos⁡x3x^2 \sin x + x^3 \cos x
Answer
3x2sin⁡x+x3cos⁡x3x^2 \sin x + x^3 \cos x

Find the derivative of (2x + 1)^5

ddx (2x+1)5\frac{d}{dx}\,(2x + 1)^5
  1. A function inside a power, so chain rule
    ddxf(g(x))=f′(g(x))⋅g′(x)\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)
  2. Differentiate the outside, inside untouched
    5(2x+1)45(2x + 1)^4
  3. Multiply by the derivative of the inside
    5(2x+1)4⋅25(2x + 1)^4 \cdot 2
Answer
10(2x+1)410(2x + 1)^4

Find the derivative of (x^2 + 1)/(x - 3)

ddx(x2+1x−3)\frac{d}{dx}\left( \frac{x^2 + 1}{x - 3} \right)
  1. Quotient rule
    (uv)′=u′v−uv′v2\left(\frac{u}{v}\right)' = \frac{u'v - uv'}{v^2}
  2. Identify the parts
    u=x2+1,  u′=2x,v=x−3,  v′=1u = x^2 + 1,\; u' = 2x, \qquad v = x - 3,\; v' = 1
  3. Substitute
    2x(x−3)−(x2+1)(1)(x−3)2\frac{2x(x - 3) - (x^2 + 1)(1)}{(x - 3)^2}
  4. Expand the numerator
    2x2−6x−x2−1(x−3)2\frac{2x^2 - 6x - x^2 - 1}{(x - 3)^2}
  5. Collect
    x2−6x−1(x−3)2\frac{x^2 - 6x - 1}{(x - 3)^2}
Answer
x2−6x−1(x−3)2\dfrac{x^2 - 6x - 1}{(x - 3)^2}

Find dy/dx by implicit differentiation for x^2 + y^2 = 25

x2+y2=25x^2 + y^2 = 25
  1. Differentiate both sides with respect to x
    2x+2ydydx=02x + 2y \frac{dy}{dx} = 0
  2. Every y term carries a dy/dx
    ddx(y2)=2ydydx\frac{d}{dx}(y^2) = 2y \frac{dy}{dx}
  3. Isolate dy/dx
    2ydydx=−2x2y \frac{dy}{dx} = -2x
  4. Divide
    dydx=−xy\frac{dy}{dx} = -\frac{x}{y}
Answer
dydx=−xy\dfrac{dy}{dx} = -\dfrac{x}{y}

Where marks get lost

Forgetting the inside derivative

The derivative of sin(3x) is 3cos(3x), not cos(3x). The chain rule’s multiplier is the most commonly dropped factor in calculus.

Reversing the quotient rule

Bottom times derivative of top comes first. Swapping the order flips the sign of the whole answer.

Using the power rule on an exponential

The derivative of 2ˣ is 2ˣ ln 2, not x·2ˣ⁻¹. The power rule needs the variable in the base, not the exponent.

Questions

Can it do second and higher derivatives?

Yes. Ask for the second derivative, or the nth, and it differentiates repeatedly, showing each round.

Does it do partial derivatives?

Yes. Name the variable, for example "partial derivative with respect to y", and the others are held constant.

Can it find the tangent line?

Yes. Ask for the tangent at a point and it differentiates, evaluates the slope there, and writes the line in point-slope form.

How do I type a derivative?

Plain text is fine: "derivative of x^3 sin x", "d/dx (2x+1)^5", or just paste the function. The symbol bar has d/dx, ∫ and lim if you want them.