Find the derivative of y = [A(x)]^α[B(x)]^β[C(x)]^γ , where α, β, and γ are constants and A, B, and C are positive functions.

Question

Find the derivative of [latex]y = [A(x)]^{α}[B(x)]^{β}[C(x)]^{γ} ,[/latex] where [latex]α, β, ext{and} γ[/latex] are constants and A, B, and C are positive functions.

Accepted Answer

First, take the natural logarithm of each side to obtain

[latex]\ln y=\alpha\ln(A(x))+\beta\ln(B(x))+\gamma\ln(C(x))[/latex]

Differentiation w.r.t. x yields

[latex]{\frac{y^{\prime}}{y}}=\alpha{\frac{A^{\prime}(x)}{A(x)}}+\beta{\frac{B^{\prime}(x)}{B(x)}}+\gamma{\frac{C^{\prime}(x)}{C(x)}}[/latex]

Multiplying by y, we have

[latex]y^{\prime}=\left[\alpha\frac{A^{\prime}(x)}{A(x)}+\beta\frac{B^{\prime}(x)}{B(x)}+\gamma\frac{C^{\prime}(x)}{C(x)}\right][A(x)]^{\alpha}[B(x)]^{\beta}[C(x)]^{\gamma}[/latex]

Question 6.11.5: Find the derivative of y = [A(x)]^α[B(x)]^β[C(x)]^γ , where ......

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