Problem

(b) Let $A(x)=(x-2)^{2}(3 x-4)^{2}$ Find $A^{\prime}(x)$ in fully factorised form

Answer

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Answer

\(\boxed{A'(x) = (x-2)^{2}(18x-24) + 2(x-2)(3x-4)^{2}}\)

Steps

Step 1 ：Let \(A(x) = (x-2)^{2}(3x-4)^{2}\)

Step 2 ：Find the derivatives of \((x-2)^{2}\) and \((3x-4)^{2}\) using the chain rule:

Step 3 ：\(\frac{d}{dx}(x-2)^{2} = 2(x-2)\)

Step 4 ：\(\frac{d}{dx}(3x-4)^{2} = 2(3x-4)(3)\)

Step 5 ：Apply the product rule to find \(A'(x)\):

Step 6 ：\(A'(x) = (x-2)^{2}(2(3x-4)(3)) + (2(x-2))(3x-4)^{2}\)

Step 7 ：Simplify the expression:

Step 8 ：\(A'(x) = (x-2)^{2}(6(3x-4)) + 2(x-2)(3x-4)^{2}\)

Step 9 ：\(\boxed{A'(x) = (x-2)^{2}(18x-24) + 2(x-2)(3x-4)^{2}}\)

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