Derivatives of exponential and logarithmic functions
Money in a bank grows faster when it is bigger: its rate of growth is proportional to its size. The exponential function eˣ is the one whose rate of change equals itself.
d/dx eˣ = eˣ; d/dx aˣ = aˣ ln a (a > 0); d/dx ln x = 1/x (x > 0); d/dx log_a x = 1/(x ln a). Use the chain rule for functions of functions.
Formula: d/dx eˣ = eˣ, d/dx ln x = 1/x
Rule 1: Differentiate exponential and logarithmic forms — Use the chain rule for the inside.
- Easy: d/dx e^(3x) = 3 e^(3x).
- Medium: d/dx ln(2x + 1) = 2 / (2x + 1).
- Exam-type: d/dx [e^(x²) sin x] = e^(x²) (2x sin x + cos x).
Rule 2: Combine with products — Name u and v, then apply the product rule.
- Easy: d/dx 5ˣ = 5ˣ ln 5.
- Medium: d/dx (x² ln x) = 2x ln x + x.
- Exam-type: d/dx ln(sin x) = cot x.
The derivative of ln of something is (derivative of the something) / (the something).
Writing d/dx aˣ = x aˣ⁻¹ (that is the power rule, not the exponential rule).
eˣ never changes under d/dx; ln x gives 1 over x.
Practice questions
- 1Differentiate e^(5x).
- 2Differentiate ln(x²).
- 3Differentiate 3ˣ.
- 4Differentiate e^(−x²).
- 5Differentiate ln(x² + 1).
- 6Differentiate x ln x.
- 7Differentiate e^(sin x).
- 8Differentiate ln(sec x + tan x).
Show answers (try them yourself first)
- 15 e^(5x)
- 22x / x² = 2 / x
- 33ˣ ln 3
- 4−2x e^(−x²)
- 52x / (x² + 1)
- 6ln x + 1
- 7cos x · e^(sin x)
- 8(sec x tan x + sec²x) / (sec x + tan x) = sec x
Logarithmic differentiation
Multiplying many factors is hard to differentiate. Taking logarithms turns the product into a sum, which is much easier.
For y = u^v (variable power) or a long product or quotient, take ln of both sides, differentiate using the chain rule, then multiply by y: y′ = y × (derivative of ln y).
Formula: y′ = y × d/dx (ln y)
Rule 1: Variable power — Write ln y = v ln u and differentiate.
- Easy: y = x^x gives ln y = x ln x, so y′ = x^x (1 + ln x).
- Medium: y = (sin x)^x gives y′ = (sin x)^x (ln sin x + x cot x).
- Exam-type: y = x^(sin x) gives y′ = x^(sin x) (cos x ln x + sin x / x).
Rule 2: Long products and quotients — Differentiate the sum of logarithms.
- Easy: y = (x + 1)²(x + 2)³ gives y′ = y [2/(x + 1) + 3/(x + 2)].
- Medium: y = x² √(x + 1) / (x + 3) gives y′ = y [2/x + 1/(2(x + 1)) − 1/(x + 3)].
- Exam-type: y = 2^(x²) gives ln y = x² ln 2, so y′ = 2x ln 2 · 2^(x²).
After differentiating ln y, always multiply back by y.
Forgetting to multiply by y at the end.
Take the log, tame the product, multiply by y.
Practice questions
- 1Differentiate y = x^(2x).
- 2Differentiate y = (cos x)^x.
- 3Differentiate y = x^(x²).
- 4Differentiate y = (x + 2)³ (x − 1)².
- 5Differentiate y = x^(1/x).
- 6Differentiate y = eˣ x³ by taking logarithms.
- 7Differentiate y = √((x − 1) / (x + 1)).
- 8Differentiate y = x^x + 2ˣ.
Show answers (try them yourself first)
- 1ln y = 2x ln x; y′ = 2 x^(2x) (ln x + 1)
- 2y′ = (cos x)^x (ln cos x − x tan x)
- 3y′ = x^(x²) · x (2 ln x + 1)
- 4y′ = y [3 / (x + 2) + 2 / (x − 1)]
- 5y′ = x^(1/x) (1 − ln x) / x²
- 6y′ = eˣ x³ (1 + 3/x)
- 7y′ = y / 2 × [1/(x − 1) − 1/(x + 1)] = y / (x² − 1)
- 8y′ = x^x (1 + ln x) + 2ˣ ln 2