Mathematics

Mathematics

Problem

Let y=[(x2+1)(x−2)3]xy=\big[(x^{2}+1)(x-2)^{3}\big]^{\sqrt{x}} for x>2x>2. a. (i) Obtain dydx\dfrac{dy}{dx}, simplifying the result as far as possible. b. (ii) Using your answer to part (i), find the numerical value of dydx\dfrac{dy}{dx} when x=3x=3. (Give your answer correct to three significant figures.)
Intuitive explanation

The key is to recognize that a variable exponent like [⋯ ]x[\cdots]^{\sqrt{x}} demands logarithmic differentiation—you can't use the power rule directly. Take the natural log of both sides to bring the exponent down, then differentiate implicitly. The product rule and chain rule will handle the base (x2+1)(x−2)3(x^2+1)(x-2)^3, while the product rule manages the x\sqrt{x} multiplied by that logarithm.

Intuitive explanation

The key is to recognize that a variable exponent like [⋯ ]x[\cdots]^{\sqrt{x}} demands logarithmic differentiation—you can't use the power rule directly. Take the natural log of both sides to bring the exponent down, then differentiate implicitly. The product rule and chain rule will handle the base (x2+1)(x−2)3(x^2+1)(x-2)^3, while the product rule manages the x\sqrt{x} multiplied by that logarithm.

Browse the problem library