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Derivative β€” Definition, Formula & Examples

Derivative

A function which gives the slope of a curve; that is, the slope of the line tangent to a function. The derivative of a function f at a point x is commonly written f '(x). For example, if f(x) = x3 then f '(x) = 3x2. The slope of the tangent line when x = 5 is f '(x) = 3·52 = 75.

Definitions of Derivatives:fβ€²(a)=lim⁑xβ†’af(x)βˆ’f(a)xβˆ’afβ€²(x)=lim⁑hβ†’0f(x+h)βˆ’f(x)hdydx=lim⁑Δxβ†’0Ξ”yΞ”xf'(a) = \lim_{x \to a}\frac{f(x) - f(a)}{x - a} \qquad f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h} \qquad \frac{dy}{dx} = \lim_{\Delta x \to 0}\frac{\Delta y}{\Delta x}Some of the notations for the derivative of y=f(x):y = f(x)\text{:}fβ€²(x)fβ€²dfdxddxf(x)Dfyβ€²(x)yβ€²dydxddxyDy\begin{array}{lllll} f'(x) & f' & \dfrac{df}{dx} & \dfrac{d}{dx}f(x) & Df \\[1em] y'(x) & y' & \dfrac{dy}{dx} & \dfrac{d}{dx}y & Dy \end{array}

See also

Difference quotient, limit, derivative rules, differentiable, differentiation, continuously differentiable

Key Formula

fβ€²(x)=lim⁑hβ†’0f(x+h)βˆ’f(x)hf'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}
Where:
  • f(x)f(x) = The original function whose derivative you want to find
  • fβ€²(x)f'(x) = The derivative of f, giving the slope of the tangent line at any point x
  • hh = A small increment that approaches zero
  • f(x+h)f(x + h) = The value of the function at a point slightly ahead of x

Worked Example

Problem: Find the derivative of f(x) = xΒ² using the limit definition, then evaluate it at x = 3.
Step 1: Write the difference quotient by substituting f(x) = xΒ² into the limit definition.
fβ€²(x)=lim⁑hβ†’0(x+h)2βˆ’x2hf'(x) = \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h}
Step 2: Expand (x + h)Β² and simplify the numerator.
=lim⁑hβ†’0x2+2xh+h2βˆ’x2h=lim⁑hβ†’02xh+h2h= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} = \lim_{h \to 0} \frac{2xh + h^2}{h}
Step 3: Factor h from the numerator and cancel with the denominator.
=lim⁑hβ†’0h(2x+h)h=lim⁑hβ†’0(2x+h)= \lim_{h \to 0} \frac{h(2x + h)}{h} = \lim_{h \to 0} (2x + h)
Step 4: Let h approach 0 to find the derivative function.
fβ€²(x)=2xf'(x) = 2x
Step 5: Evaluate the derivative at x = 3 to find the slope of the tangent line at that point.
fβ€²(3)=2(3)=6f'(3) = 2(3) = 6
Answer: The derivative of f(x) = xΒ² is f'(x) = 2x. At x = 3, the slope of the tangent line is 6.

Another Example

This example uses the power rule shortcut instead of the limit definition, showing how derivative rules speed up the process for polynomial functions.

Problem: Find the derivative of f(x) = 5xΒ³ βˆ’ 4x + 7 using the power rule, then find the slope of the tangent line at x = 2.
Step 1: Apply the power rule to each term. The power rule states that the derivative of xⁿ is nxⁿ⁻¹. Constants have a derivative of 0.
ddx[xn]=nxnβˆ’1\frac{d}{dx}[x^n] = n x^{n-1}
Step 2: Differentiate the first term: 5xΒ³ becomes 5 Β· 3xΒ².
ddx[5x3]=15x2\frac{d}{dx}[5x^3] = 15x^2
Step 3: Differentiate βˆ’4x (which is βˆ’4xΒΉ) and the constant 7.
ddx[βˆ’4x]=βˆ’4,ddx[7]=0\frac{d}{dx}[-4x] = -4, \quad \frac{d}{dx}[7] = 0
Step 4: Combine the results and evaluate at x = 2.
fβ€²(x)=15x2βˆ’4β‡’fβ€²(2)=15(4)βˆ’4=56f'(x) = 15x^2 - 4 \quad \Rightarrow \quad f'(2) = 15(4) - 4 = 56
Answer: f'(x) = 15xΒ² βˆ’ 4, and the slope of the tangent line at x = 2 is 56.

Frequently Asked Questions

What is the difference between a derivative and a slope?
Slope usually refers to the constant rate of change of a straight line (rise over run). The derivative generalizes this idea to curves: it gives the slope of the tangent line at each individual point. For a linear function like f(x) = 3x + 2, the derivative is simply the constant slope, 3. For curves, the derivative changes depending on where you are on the graph.
When do you use the limit definition versus derivative rules?
The limit definition is the foundational formula that proves what the derivative is from first principles. In practice, you use derivative rules (power rule, product rule, chain rule, etc.) because they are much faster. Most courses require you to use the limit definition a few times to understand the concept, then switch to rules for efficiency.
What does it mean when the derivative equals zero?
When f'(x) = 0 at a point, the tangent line at that point is horizontal. This means the function has momentarily stopped increasing or decreasing. These points are called critical points and often correspond to local maxima, local minima, or inflection points of the function. Finding where the derivative equals zero is a key step in optimization problems.

Derivative vs. Integral

DerivativeIntegral
DefinitionRate of change (slope) of a function at a pointAccumulation of area under a curve
Notationf'(x) or dy/dx∫ f(x) dx or F(x)
Geometric meaningSlope of the tangent lineArea between the curve and the x-axis
OperationDifferentiation β€” reduces polynomial degree by 1Integration β€” increases polynomial degree by 1
RelationshipInverse of integrationInverse of differentiation (by the Fundamental Theorem of Calculus)

Why It Matters

The derivative is the central concept of differential calculus and appears throughout science, engineering, and economics. In physics, velocity is the derivative of position with respect to time, and acceleration is the derivative of velocity. Students encounter derivatives heavily in AP Calculus, college math courses, and any field that studies how quantities change.

Common Mistakes

Mistake: Forgetting to subtract 1 from the exponent when using the power rule.
Correction: The power rule requires both multiplying by the exponent AND reducing it by 1. For x⁴, the derivative is 4x³, not 4x⁴.
Mistake: Treating the derivative of a product as the product of the derivatives: writing (fg)' = f' Β· g'.
Correction: The correct product rule is (fg)' = f'g + fg'. You must apply the product rule whenever two non-constant functions are multiplied together.

Related Terms