Logarithmic differentiation: A method in which you take the log of both sides before differentiating. It is used when powers, products, or quotients are complicated.
Example: For y = x^x
log y = x·log x → (1/y)·y' = log x + 1 → y' = x^x(log x + 1)
Differentiating an inverse function: For the inverse function x = g(y) of y = f(x),
dx/dy = 1/(dy/dx)
Example ① Find y' for y = (sin x)^x.
Solution: log y = x·log(sin x)
y'/y = log(sin x) + x·cos x/sin x = log(sin x) + x·cot x
y' = (sin x)^x · {log(sin x) + x·cot x}
Solution: log y = x·log(sin x)
y'/y = log(sin x) + x·cos x/sin x = log(sin x) + x·cot x
y' = (sin x)^x · {log(sin x) + x·cot x}
Key Point
- Logarithmic differentiation is the first choice when the base or exponent contains a variable
- After calculating, substitute y with the original expression
Practice Problem
- Find y' for y = x^{cos x}
Answer and Explanation
- Answer:x^{cos x}·{−sin x·log x + cos x/x}
log y = cos x·log x → y'/y = −sin x·log x + cos x/x.