+0

# [Veteran - member] Need help with a der. rule

0
205
1

I need a quick refresher for this, i have in 4 hours and would like to see the rules in action in a simple problem like this.

Nov 14, 2017

#1
0

Find the derivative of the following via implicit differentiation:
d/dy(g(y)) = d/dy(16 (1/y - sqrt(y) + 2 y^3))

The derivative of g(y) is g'(y):
g'(y) = d/dy(16 (1/y - sqrt(y) + 2 y^3))

Factor out constants:
g'(y) = 16 d/dy(1/y - sqrt(y) + 2 y^3)

Differentiate the sum term by term and factor out constants:
g'(y) = 16 d/dy(1/y) - d/dy(sqrt(y)) + 2 d/dy(y^3)

Use the power rule, d/dy(y^n) = n y^(n - 1), where n = -1: d/dy(1/y) = d/dy(y^(-1)) = -y^(-2):
g'(y) = 16 (-(d/dy(sqrt(y))) + 2 (d/dy(y^3)) + (-1)/(y^2))

Use the power rule, d/dy(y^n) = n y^(n - 1), where n = 1/2: d/dy(sqrt(y)) = d/dy(y^(1/2)) = y^(-1/2)/2:
g'(y) = 16 (-1/y^2 + 2 (d/dy(y^3)) - 1/(2 sqrt(y)))

Use the power rule, d/dy(y^n) = n y^(n - 1), where n = 3: d/dy(y^3) = 3 y^2:
g'(y) = 16 (-1/y^2 - 1/(2 sqrt(y)) + 2 3 y^2)

Simplify the expression:
g'(y) = 16 (-1/y^2 - 1/(2 sqrt(y)) + 6 y^2) - Courtesy of "Mathematica 11 Home Edition"

Nov 14, 2017