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Determine the derivative of a function g in terms of a function f

by RoRi

Determine the derivative of the function g in terms of the function f for the following definitions of g:

  1. g(x) = f(x^2);
  2. g(x) = f(\sin^2 x) + f(\cos^2 x);
  3. g(x) = f(f(x));
  4. g(x) = f(f(f(x))).
  1. We use the chain rule to compute,

        \[g'(x) = (x^2)' f'(x^2) = 2x f'(x^2).\]

  2. We use the chain rule to compute,

        \begin{align*} g'(x) &= (\sin^2 x)' f'(\sin^2 x) + (\cos^2 x)' f'(\cos^2 x) \\ &= 2 \sin x \cos x f'(\sin^2 x) - 2 \sin x \cos x f'(\cos^2 x) \\ &= \sin (2x) f'(\sin^2 x) - \sin (2x) f'(\cos^2 x) \\ & = \sin (2x) (f'(\sin^2 x) - f'(\cos^2 x)). \end{align*}

  3. We use the chain rule to compute,

        \[ g'(x) = (f'(f(x)))(f'(x)). \]

  4. Finally we use part (c) and the chain rule to compute,

        \[ g'(x) = f'(f(f(x))) f'(x) f'(f(x)) = f'(x) f'(f(x)) f'(f(f(x))). \]

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