Outcome A3: Derivative Rules

Question 1

Product rule

Answer 1

d/dx[f(x)g(x)] = g(x)f’(x) + f(x)*g’(x)

Question 2

Quotient rule

Answer 2

d/dx[f(x)/g(x)] = [g(x)f’(x) - f(x)g’(x)] / [g(x)]^2

Question 3

d/dx(cscx)

Answer 3

-cscxcotx

Question 4

d/dx(secx)

Answer 4

secxtanx

Question 5

d/dx(cotx)

Answer 5

-csc^2x

Question 6

What is a composition of functions?

Answer 6

The output of one function becomes an input of another. That function acts as an intermediate.

Question 7

Chain rule

Answer 7

d/dx[f(g(x))] = f’(g(x))*g’(x)

Question 8

Chain rule for 3 functions

Answer 8

d/dx[f(g(h(x)))] = f’(g(h(x)))g’(h(x))h’(x)

Question 9

What are the three laws of logarithms for a,b > 0?

Answer 9

1. ln(a*b) = ln(a) + ln(b)
2. ln(a/b) = ln(a) - ln(b)
3. ln(a^r) = r*ln(a)

Question 10

Why do we use logarithmic differentiation, and when?

Answer 10

Taking the derivative for some functions can be made easier using logarithms. We can use it when we have a large number of product rules, quotient rules, and chain rules in a particular function we are trying to differentiate.

Question 11

What are the steps to using logarithmic differentiation?

Answer 11

Given y = f(x):

1. Take ln of both sides and simplify (the right hand side) using the laws of logarithms
2. Take the derivative of both sides with respect to x
3. Solve for y’