Find the derivative of the compound function F(x) = f (g(x)) at x0 = −3 in case f (u) = u³ and g(x) = 2 − x².

Question

Find the derivative of the compound function F(x) = f (g(x)) at [latex]x_{0} = −3[/latex] in case f (u) = u³ and g(x) = 2 − x².

Accepted Answer

In this case one has [latex]f^{\prime}(u) = 3u^{2}[/latex] and [latex]g^{\prime}(x) = −2x.[/latex] So according to (6.8.3),

[latex]F^{\prime}(x_{0})=f^{\prime}(u_{0})g^{\prime}(x_{0})=f^{\prime}(g(x_{0}))g^{\prime}(x_{0})[/latex] (6.8.3)

one has [latex]F^{\prime}(−3) = f^{\prime} (g(−3)) g^{\prime}(−3).[/latex] Now g(−3) = 2 − (−3)² = 2 − 9 = −7; [latex]g^{\prime}(−3) = 6;[/latex] and [latex]f^{\prime}(g(−3)) = f ^{\prime}(−7) = 3(−7)^{2} = 3 · 49 = 147. [/latex] So [latex]F^{\prime}(−3) = f ^{\prime}(g(−3)) g^{\prime}(−3) =147 · 6 = 882.[/latex]

Question 6.8.5: Find the derivative of the compound function F(x) = f (g(x))......

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