SolveMathAI

Derivative calculator

Every rule named, every substitution shown, no subscription.

Enter to solve · Shift+Enter for a new line

  • Free
  • No account
  • Every step shown
  • 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}

Solved problems

Asked here and worked through in full. Open one for every step.

  1. Find the derivative of x^3 * sin(x)
    Answer 3x2sin⁡(x)+x3cos⁡(x)3x^2 \sin(x) + x^3 \cos(x)

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

What is a derivative?

The rate at which a function is changing, and the slope of its graph at a point. If distance is the function, the derivative is speed.

How do I find a derivative?

Apply the rule that matches the shape. The power rule drops the exponent to the front and lowers it by one, so the derivative of x³ is 3x²; products, quotients and compositions have their own rules.

What is the chain rule?

For a function inside another function, differentiate the outside and multiply by the derivative of the inside. The derivative of (3x + 1)⁵ is 5(3x + 1)⁴ × 3.

What is the product rule?

For two functions multiplied together, the derivative is the first times the derivative of the second plus the second times the derivative of the first. For x³ sin x it is x³ cos x + 3x² sin x.

What is the quotient rule?

For a fraction, take the bottom times the derivative of the top, minus the top times the derivative of the bottom, all over the bottom squared.

How do I find the derivative at a point?

Differentiate first, then substitute the value. For f(x) = 2x² − 3x + 1, the derivative is 4x − 3, so f′(2) = 5.

What is implicit differentiation?

Differentiating both sides when y is not written alone, treating y as a function of x and attaching dy/dx by the chain rule each time y is differentiated.

Does it show which rule it used?

Yes. Every step is labelled with the rule by name, so you can see where the chain rule was applied rather than only the final expression.