Fix the mistake: Differentiation ② (Composite Functions, Trigonometric Functions, Exponential and Logarithmic Functio

Differentiating Composite Functions ①: The Chain Rule

The derivative of a composite function consisting of two functions, such as y = f(g(x)), is
　{f(g(x))}′ = f'(g(x))·g'(x)　（chain rule）

Remember it as “differentiate the outer function × differentiate the inner function.”

Example ① Find y' for y = (3x+1)⁵.
Solution: Outer function = u⁵ → 5u⁴; inner function = 3x+1 → 3.
y' = 5(3x+1)⁴ · 3 = 15(3x+1)⁴
Example ② Find y' for y = √(x²+1).
Solution: (u^{1/2})' = u^{−1/2}/2; (x²+1)' = 2x.
y' = (1/(2√(x²+1)))·2x = x/√(x²+1)
Key Point
  • “Differentiate the outside × differentiate the inside”
  • The most important formula, used with powers, √, exponential functions, and logarithms

Practice Problems

  1. Find y' for y = (2x−5)⁴
  2. Find y' for y = √(3x+2)

Answers and Explanations

  1. Answer:8(2x−5)³
  2. Answer:3/(2√(3x+2))

Math (Math III): related lessons