For next lecture

MAT137Y1 – LEC0501

Calculus!

Trig derivatives

& implicit differentiation

October 29 ^th , 2018

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Oct 29, 2018 1

For next lecture

For Wednesday (Oct 31), watch the videos:

• Derivatives of exponentials and logarithms: 3.13, (3.14), 3.15, 3.16, 3.17 , 3.18

• Related rates: 3.19, 3.20, 3.21

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Oct 29, 2018 2

Derivative from a graph

Below is the graph of a function 𝑓 . Sketch the graph of its derivative 𝑓 ^′ .

−4 −3 −2 −1 1 2 3 4

−3

−2

−1 1 2 3

𝑂

𝑦 = 𝑓 (𝑥)

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Oct 29, 2018 3

Derivative of cos

Let 𝑔(𝑥) = cos(𝑥).

Obtain and prove a formula for its derivative directly from the definition of derivative as a limit.

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Oct 29, 2018 4

Derivatives of the other trig functions

Using all the basic differentiation rules, as well as sin ^′ (𝑥) = cos(𝑥), cos ^′ (𝑥) = − sin(𝑥),

quickly obtain and prove formulas for the derivatives of

1 tan ² cot ³ sec ⁴ csc

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Oct 29, 2018 5

Implicit differentiation

The equation sin(𝑥 + 𝑦) + 𝑥𝑦 ² = 0 defines a function 𝑦 = ℎ(𝑥) near (0, 0). Using implicit differentiation, compute

1 ℎ(0) ² ℎ ^′ (0) ³ ℎ ^″ (0) ⁴ ℎ ^‴ (0)

𝑥 𝑦

Jean-Baptiste Campesato MAT137Y1 – LEC0501 – Calculus! – Oct 29, 2018 6