3.6 The Chain Rule Y= f(g(x)) Y’=f’(g(x)) g’(x)

Chain rule Unlocks many derivatives

actual computations r=10, A=100 pi cm^2 r= 10.1 A= 102 pi cm^2

Use the chain rule: Y=sec(6x-3) Y= (6x-3)^7

3.7 Implicit differentiation Use when the function is not given as y in terms of x,

Or when one variable cannot be conveniently separated from the other.

3.8 Linear approximation and differentials For x values close to “x “ the tangent line approximates the function.

Equation of the tangent line at the point ( c, f (c) ) y= f(c) + f’ (c) (x-c)

Differentials “ dx” can have a numeric value

Example How is the area of a circle changing when the radius is increasing at 0.01 cm/sec at the instant the radius is 5 cm ?

Using differentials to approximate the increase in a circle’s area Radius = 10 cm Radius is increased by 0.1 cm

Ex. The edges of a cube are increasing at 3 cm/sec. How fast is the volume changing when: a) edge= 1 cm b) edge = 10 cm

Approximate the cube root of 8.1 using differentials.

Approximate the square root of 62.

Volume of a fluid flowing through a pipe at a fixed pressure v=kr^4 How will a 10 % increase in the radius affect the volume?

Find the differential – dy – using implicit differentiation xy^2 +x^2 y – 4 = 0

Review problems

Find the amount of possible error in computing the area Circle Diameter=200 cm Diameter measured with cm

Find the amount of possible error in computing the area Equilateral triangle Side = 90 m Side measured with + 1 m

Use differentials to find the cube root of 64.12

Test on all of chapter 3 Hint Know everything