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Determine the derivative of a function satisfying f(x2) = x3

by RoRi

Given a function defined for all positive real numbers which satisfies a functional equation

    \[ f(x^2) = x^3 \]

for all positive real numbers, compute f'(4).

We take the derivatives of both sides, using the chain rule to take the derivative of f(x^2):

    \begin{align*} f(x^2) = x^3 && \implies && (f(x^2))' &= (x^3)' \\ && \implies && 2x f'(x^2) &= 3x^2 \\ && \implies && f'(x^2) &= \frac{3}{2}x. \end{align*}

(We can divide by x since by assumption x is a positive real number; hence, not zero.) Then we have

    \[ f'(4) = f'(2^2) = \frac{3}{2} (2) = 3. \]

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