{The Derivative

Slope of a line, secant line, tangent line, slope of a secant line, slope of a line tangent to a curve

Lines, slope, point-slope form

What do we know about slopes of lines?

Lines, slope, point-slope form

Lines, slope, point-slope form

Lines, slope, point-slope form

Lines, slope, point-slope form

Given a curve, the line that intersects the curve on two points is called a secant line.

Secant line

Given a curve, the line that intersects the curve on two points is called a secant line.

Secant line

Tangent line

Given a non-linear curve, how do you get the slope of the curve at a the point (x, f(x) )?

Challenge question

Given a non-linear curve, how do you get the slope of the curve at a the point (x, f(x) )?

Challenge question

Let be the slope of the tangent line, then the slope of the curve at (x,f(x)) is the same as is the derivative of f at x. 𝑓 ′ (𝑥 )=𝑚𝑇𝐿

𝑦= 𝑓 (𝑥 )

How do we compute

Challenge question

How to find

(𝑥 𝑖 , 𝑓 (𝑥 𝑖 ) ) 𝑚𝑇𝐿=Δ𝑦Δ𝑥

=𝑦2− 𝑦1𝑥2−𝑥2

𝑓 ′ (𝑥 )= limΔ𝑥→ 0

𝑓 (𝑥+∆ 𝑥 )− 𝑓 (𝑥 )∆𝑥

Three step rule1. Find 2. Find

3. Evaluate

How to compute for

limΔ𝑥→0

𝑓 (𝑥+∆𝑥 )− 𝑓 (𝑥 )∆ 𝑥

How to compute for

First step

How to compute for

Second step

How to compute for

Third step

𝑓 ′ (𝑥 )=4 𝑥−3

What is

𝑓 ′ (2 )=4 (2 )−3=5

What is

𝑓 ′ (0 )=4 (0 )−3=−3