Derivative Calculator

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∑Math Input
x^3 + 2x^2 - 5x
sin(x) * cos(x)
e^(2x)
ln(x^2 + 1)

What is a Derivative?

A derivative measures the instantaneous rate of change of a function. For a function f(x)f(x), the derivative f′(x)f'(x) is defined as:

f′(x)=lim⁡h→0f(x+h)−f(x)hf'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Geometrically, the derivative at a point equals the slope of the tangent line to the function's graph at that point.

Common notations:

  • f′(x)f'(x) — Lagrange notation
  • dydx\frac{dy}{dx} — Leibniz notation
  • y˙\dot{y} — Newton notation (used in physics)

Basic Derivative Rules

Power Rule

ddxxn=nxn−1\frac{d}{dx} x^n = nx^{n-1}

Sum / Difference Rule

ddx[f(x)±g(x)]=f′(x)±g′(x)\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

Product Rule

ddx[f(x)⋅g(x)]=f′(x)g(x)+f(x)g′(x)\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)

Quotient Rule

ddx[f(x)g(x)]=f′(x)g(x)−f(x)g′(x)[g(x)]2\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}

Chain Rule

ddx[f(g(x))]=f′(g(x))⋅g′(x)\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)

Common Derivatives

FunctionDerivative
sin⁡x\sin xcos⁡x\cos x
cos⁡x\cos x−sin⁡x-\sin x
exe^xexe^x
ln⁡x\ln x1x\frac{1}{x}
axa^xaxln⁡aa^x \ln a

Common Mistakes to Avoid

  • Forgetting the chain rule: When differentiating composite functions like sin⁡(3x)\sin(3x), don't forget to multiply by the inner derivative (33).
  • Power rule sign errors: ddxx−2=−2x−3\frac{d}{dx} x^{-2} = -2x^{-3}, not −2x−1-2x^{-1}.
  • Confusing product and chain rules: (fg)′=f′g+fg′(fg)' = f'g + fg' is the product rule; (f∘g)′=f′(g)⋅g′(f \circ g)' = f'(g) \cdot g' is the chain rule.
  • Forgetting constants: The derivative of a constant is 00, not 11.

Examples

Step 1: Apply power rule to each term: ddx(x3)=3x2\frac{d}{dx}(x^3) = 3x^2
Step 2: ddx(2x2)=4x\frac{d}{dx}(2x^2) = 4x, ddx(−5x)=−5\frac{d}{dx}(-5x) = -5, ddx(3)=0\frac{d}{dx}(3) = 0
Step 3: Combine: f′(x)=3x2+4x−5f'(x) = 3x^2 + 4x - 5
Answer: f′(x)=3x2+4x−5f'(x) = 3x^2 + 4x - 5

Step 1: Apply product rule: f′(x)=cos⁡(x)⋅cos⁡(x)+sin⁡(x)⋅(−sin⁡(x))f'(x) = \cos(x) \cdot \cos(x) + \sin(x) \cdot (-\sin(x))
Step 2: Simplify: f′(x)=cos⁡2(x)−sin⁡2(x)=cos⁡(2x)f'(x) = \cos^2(x) - \sin^2(x) = \cos(2x)
Answer: f′(x)=cos⁡(2x)f'(x) = \cos(2x)

Step 1: Apply chain rule: outer function eue^u where u=2xu = 2x
Step 2: f′(x)=e2x⋅ddx(2x)=e2x⋅2f'(x) = e^{2x} \cdot \frac{d}{dx}(2x) = e^{2x} \cdot 2
Answer: f′(x)=2e2xf'(x) = 2e^{2x}

Frequently Asked Questions

The power rule states that the derivative of x^n is n·x^(n-1). For example, the derivative of x³ is 3x².

Use the chain rule when differentiating composite functions — functions inside other functions, like sin(3x), e^(x²), or ln(2x+1). Multiply the outer derivative by the inner derivative.

A derivative finds the rate of change (slope) of a function, while an integral finds the accumulated area under a curve. They are inverse operations of each other.

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