Question

Given \( f(x) = x - 1 \), \( h(1) = 4 \), and \( h'(1) = -2 \). If \( g(x) = (f[4(h(x)) + 3])^2 \), find the value of \( g'(1) \).

• A. \( -288 \)
• B. \( 288 \)
• C. \( -144 \)
• D. \( 144 \)

Hint:

Whenever a question contains nested functions like \( (f(g(h(x))))^n \), always differentiate layer-by-layer from outside to inside. Simplifying the given function before differentiating saves a lot of time and reduces calculation mistakes.

Answer 1

The Correct Option is A

Concept: This problem involves differentiation of a composite function. Whenever a function is present inside another function, we use the Chain Rule. The chain rule states: \[ \frac{d}{dx}[F(G(x))] = F'(G(x))\cdot G'(x) \] If the outer function is squared, then differentiation proceeds from the outermost layer to the innermost layer step-by-step.

Step 1: Simplifying the given function first.
We are given: \[ f(x)=x-1 \] and \[ g(x)=\left(f[4h(x)+3]\right)^2 \] First evaluate the expression inside \(f\): \[ f[4h(x)+3]=(4h(x)+3)-1 \] since \(f(t)=t-1\). Therefore, \[ f[4h(x)+3]=4h(x)+2 \] Hence the function becomes: \[ g(x)=(4h(x)+2)^2 \] So the complicated composite expression is now reduced into a simple squared function.

Step 2: Applying the Chain Rule carefully.
Differentiate: \[ g(x)=(4h(x)+2)^2 \] Using: \[ \frac{d}{dx}(u^2)=2u\frac{du}{dx} \] where \[ u=4h(x)+2 \] Thus, \[ g'(x)=2(4h(x)+2)\cdot \frac{d}{dx}(4h(x)+2) \] Now differentiate the inner term: \[ \frac{d}{dx}(4h(x)+2)=4h'(x) \] Substituting: \[ g'(x)=2(4h(x)+2)\cdot 4h'(x) \] Therefore, \[ g'(x)=8(4h(x)+2)h'(x) \]

Step 3: Substituting the given values.
We are given: \[ h(1)=4,\qquad h'(1)=-2 \] Substitute into the derivative: \[ g'(1)=8(4(4)+2)(-2) \] Simplify inside brackets: \[ 4(4)=16 \] so, \[ 16+2=18 \] Hence, \[ g'(1)=8(18)(-2) \] Now, \[ 8\times18=144 \] Therefore, \[ g'(1)=144(-2)=-288 \] Thus, \[ \boxed{-288} \]