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Derivative Calculator

Read asf(x)=x3−4 x+1f(x) = x^{3} - 4\,x + 1

Powers ^ · multiply with * or a space · sqrt( ) ln( ) log( ) sin( ) cos( ) tan( ) e pi |x| · implicit products like 2x or x sin x work

Try an example

Derivative

Accepts numbers and constants such as pi/2 or e

Derivative

f′(x)=3 x2−4f'(x) = 3\,x^{2} - 4
3x^2 - 4

At x = 2

f(2)=1f(2) = 1
f′(2)=8f'(2) = 8

The function is increasing at this point: near x = 2, f changes by about 8 for every unit step in x.

Graph

-2-10123-5051015202530
f(x)f′(x)

Where f′(x) = 0 (searched from x = -8 to 12)

  • x ≈ -1.1547, f ≈ 4.0792: local maximum
  • x ≈ 1.1547, f ≈ -2.0792: local minimum

Term by term

ddx[x3]=3 x2\frac{d}{dx}\left[x^{3}\right] = 3\,x^{2}
ddx[−4 x]=−4\frac{d}{dx}\left[-4\,x\right] = -4
ddx[1]=0\frac{d}{dx}\left[1\right] = 0

Rules used

  • Sum and difference rule(u±v)′=u′±v′(u \pm v)' = u' \pm v'
  • Power ruleddx xn=n xn−1\frac{d}{dx}\,x^{n} = n\,x^{n-1}
  • Constant multiple rule(c u)′=c u′(c\,u)' = c\,u'

Differentiate a function of x symbolically using the power, product, quotient and chain rules. Take the first, second or third derivative, evaluate it at a point, and graph f and f′ together to see where the function rises, falls and turns.

Formula

f′(x)=lim⁡h→0(f(x+h)−f(x))/h\text{f}'(\text{x}) = \lim_{\text{h} \to 0} (\text{f}(\text{x} + \text{h}) - \text{f}(\text{x})) / \text{h}
(uv)′=u′v+uv′(\text{uv})' = \text{u}'\text{v} + \text{uv}'
(f(g(x)))′=f′(g(x))×g′(x)(\text{f}(\text{g}(\text{x})))' = \text{f}'(\text{g}(\text{x})) \times \text{g}'(\text{x})

Worked Example

f(x)x² sin x
Evaluate atx = π
Result
f′(x)2x sin x + x² cos x
f′(π)−π² ≈ −9.87

By the product rule with u = x² and v = sin x, f′ = 2x · sin x + x² · cos x. At x = π, sin π = 0 and cos π = −1, so f′(π) = −π²: the function is falling steeply there.

Frequently Asked Questions

What are the basic rules of differentiation?

Power rule: (xⁿ)′ = nxⁿ⁻¹. Product rule: (uv)′ = u′v + uv′. Quotient rule: (u/v)′ = (u′v − uv′)/v². Chain rule: (f(g(x)))′ = f′(g(x)) · g′(x). Standard results: (sin x)′ = cos x, (cos x)′ = −sin x, (eˣ)′ = eˣ and (ln x)′ = 1/x.

What does the derivative tell you about a graph?

Its sign shows whether the function is increasing (f′ > 0) or decreasing (f′ < 0), and its zeros mark possible maxima and minima. The second derivative gives the concavity: f″ > 0 curves upward and f″ < 0 downward.

How do you differentiate x to the power x?

Write xˣ as e^(x ln x) and apply the chain rule: (xˣ)′ = xˣ(ln x + 1). This logarithmic differentiation works whenever both the base and the exponent depend on x.

Common Mistakes

  • !Forgetting the chain rule: the derivative of sin(3x) is 3cos(3x), not cos(3x).
  • !Using the power rule on an exponential: the derivative of 2ˣ is 2ˣ ln 2, not x·2ˣ⁻¹.

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