Question

question number 5 (4.00 points)
the function ( f ) is differentiable. given
find ( h(x) )
( h(x) = 3f(x)^4 + 4f(3x) )
( circ h(x) = 12f(x)^3 + 12f(3x) )
( circ h(x) = 3f(x)^3 cdot f(x) + 12f(3x) )
( circ h(x) = 12f(x)^3 cdot f(x) + 12f(3x) cdot f(3x) )
( circ h(x) = 12f(x)^3 cdot f(x) + 4f(3x) )
( circ h(x) = 12f(x)^3 cdot f(x) + 12f(3x) )
( circ )none of the above.

Explanation:

Step1: Differentiate \(3[f(x)]^4\)

Using the chain rule, if \(u = f(x)\), then \(\frac{d}{dx}(3u^4)=3\times4u^3\times u' = 12[f(x)]^3f'(x)\)

Step2: Differentiate \(4f(3x)\)

Using the chain rule, let \(u = 3x\), then \(\frac{d}{dx}(4f(u)) = 4f'(u)\times u' = 4f'(3x)\times3 = 12f'(3x)\)

Step3: Combine the derivatives

\(h'(x)=12[f(x)]^3f'(x)+12f'(3x)\)

Answer:

\(h'(x) = 12[f(x)]^3 \cdot f'(x) + 12f'(3x)\) (the fifth option)