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Problem 52 Hard Difficulty

If $ f $ is a differentiable function, find an expression for the derivative of each of the following functions.

(a) $ y = x^2 f(x) $

(b) $ y = \frac {f(x)}{x^2} $

(c) $ y = \frac {x^2}{f(x)} $

(d) $ y = \frac {1+2 xf(x)}{\sqrt{x}} $

Answer

(a) $\frac{d y}{d x}=x^{2} \cdot f^{\prime}(x)+2 x \cdot f(x)$
(b) $\frac{f^{\prime}(x) x-2 f(x)}{x^{3}}$
(c) $y^{\prime}=\frac{2 x f(x)-x^{2} f^{\prime}(x)}{(f(x))^{2}}$
(d) $\frac{2 x^{2} f^{\prime}(x)+x f(x)-1}{2 x^{3 / 2}}$

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Video Transcript

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