Topic 2 of 7

Definition of the derivative

Step 15 · Calculus 1 topics

See the derivative as the slope of the tangent line, compute it from the limit definition, and read it as a rate of change.

Slope of a tangent line

A secant line joins two points on a curve, and its slope is the average rate of change between them. Slide the second point toward the first and the secant turns into the tangent line, which just touches the curve there.

m_{\text{sec}} = \frac{f(a + h) - f(a)}{h}

y = x². The secant through (1, 1) and (2, 4) has slope 3. The tangent at (1, 1), dashed, has slope 2.

Example: Secant slopes

For f(x) = x², find the slope of the secant from x = 1 to x = 1 + h, for h = 1, 0.1 and 0.01.

The slope is ((1 + h)² − 1) ÷ h, which simplifies to 2 + h.
h = 1 gives 3, h = 0.1 gives 2.1, and h = 0.01 gives 2.01.

Answer3, 2.1 and 2.01, closing in on 2

The limit definition

The derivative of f at a is the limit of the secant slopes as h shrinks to 0. When this limit exists, f is differentiable at a.

f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h}

Example: A quadratic

Use the definition to find f′(x) for f(x) = x² + 3x.

f(x + h) − f(x) works out to 2xh + h² + 3h.
Divide by h: 2x + h + 3.
Let h go to 0.

Answerf′(x) = 2x + 3

Example: A reciprocal

Use the definition to find the derivative of $f(x) = \frac{1}{x}$ .

$\frac{1}{x + h} - \frac{1}{x} = \frac{-h}{x(x + h)}$ .
Divide by h: $\frac{-1}{x(x + h)}$ .
Let h go to 0.

Answer $f'(x) = -\frac{1}{x^{2}}$

The derivative as a rate

f′(a) is the instantaneous rate of change of f at a: how fast the output changes per unit of input. If s(t) is position, s′(t) is velocity. If C(n) is the cost of n items, C′(n) is roughly the cost of one more.

Example: Velocity

A ball's height is s(t) = 20t − 5t² meters after t seconds. Find its velocity at t = 1.

The same method as above gives s′(t) = 20 − 10t.
s′(1) = 20 − 10 = 10.

Answer10 m/s, upward

Example: Reading the sign

f′(3) = −2. What does f do near x = 3?

The tangent slope is negative.
So f is decreasing there, losing about 2 units of output for each unit of input.

AnswerIt is decreasing

Notation	Read as
$f'(x)$	f prime of x
$\frac{dy}{dx}$	dy by dx
$\frac{d}{dx}f(x)$	the derivative of f(x)

Try it yourself

Work each one on paper first, then open the answer to check.

Find the slope of the secant line of f(x) = x² from x = 2 to x = 5.
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7
Use the definition to find f′(x) for f(x) = 5x − 2.
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5
Use the definition to find f′(x) for f(x) = 3x².
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6x
For f(x) = x², what is f′(4)?
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8
A rock's height is s(t) = 45 − 5t² meters. What is its velocity at t = 2?
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−20 m/s: it is falling at 20 m/s
Where does f(x) = |x − 2| have no derivative?
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At x = 2, where it has a corner