3.7 Derivatives of Logarithmic Functions

Question 1

d/dx (log_a x) =

Answer 1

1 / (x lna)

Question 2

d/dx (ln x) =

Answer 2

1 / x

Question 3

d/dx (ln u) =

Answer 3

(1 / u) du/dx

Question 4

d/dx [ln g(x)] =

Answer 4

g’(x) / g(x)

Question 5

d/dx (ln |x|) =

Answer 5

1 / x

Question 6

Steps in Logarithmic Differentiation:

Answer 6

1. Take natural logarithms pf both sides of an equation y =f(x) and use the Laws of Logarithms to simplify.
2. Differentiate implicitly with respect to x.
3. Solve the resulting equation for y’

Question 7

The Power Rule

Answer 7

If n is any real number and f(x) = xⁿ, then

f’(x) = nx^n-1

Question 8

Why can’t you use the Power Rule to say that

d/dx x^u = ux^x-1

Answer 8

Because the Power Rule only applies when n is a constant, and U is not a constant, it is a function.

Question 9

e =

Answer 9

lim _{x → 0} (1 + x)^1/x

Question 10

expression for e using n

Answer 10

lim _{n → ∞} (1 + 1/n)ⁿ