Derivatives Of Log, Ln, and E^x

Question 1

What is the derivative of ln(x)?

Answer 1

1 / x

x > 0

Question 2

What is the derivative of ln(f(x))?

Answer 2

1 / f(x) * f’(x)

Chain rule

Question 3

What is the derivative of log_a(x)?

Answer 3

1 / (x * ln(a))

a > 0, a != 1

Question 4

What is the derivative of log_a(f(x))?

Answer 4

1 / (f(x) * ln(a)) * f’(x)

Chain rule

Question 5

What is the derivative of e^x?

Answer 5

e^x

Self-derivative

Question 6

What is the derivative of e^{f(x)}?

Answer 6

e^{f(x)} * f’(x)

Chain rule

Question 7

What is the derivative of a^x?

Answer 7

a^x * ln(a)

Exponential base

Question 8

What is the Product Rule for logarithms?

Answer 8

log_b(xy) = log_b x + log_b y

Question 9

What is the Quotient Rule for logarithms?

Answer 9

log_b(x/y) = log_b x - log_b y

Question 10

What is the Power Rule for logarithms?

Answer 10

log_b(x^n) = n log_b x

Question 11

What is the Change of Base formula?

Answer 11

log_b x = log_c x / log_c b

Question 12

What is the Zero Rule for logarithms?

Answer 12

log_b(1) = 0

Question 13

What is the Identity Rule for logarithms?

Answer 13

log_b(b) = 1

Question 14

What is the Equality Rule for logarithms?

Answer 14

log_b x = log_b y => x = y

Question 15

What is the Inverse Rule for logarithms?

Answer 15

b^{log_b x} = x

Question 16

What is the Reciprocal Rule for logarithms?

Answer 16

log_b(1/x) = -log_b x