Question

question 24
let ( f(x) = (12x^2 + 36)^5 ).
find the differential, ( df ).
( df = 120x(12x^2 + 36)^4 )
( df = 5(12x^2 + 36)^4 dx )
( df = (60x^2 + 36)^4 dx )
( df = 120x(12x^2 + 36)^4 dx )

Explanation:

Step1: Identify outer/inner functions

Let $u=12x^2+36$, so $f(u)=u^5$.

Step2: Apply chain rule to find $f'(x)$

$f'(x)=\frac{df}{du}\cdot\frac{du}{dx}=5u^4\cdot(24x)$
Substitute back $u=12x^2+36$:
$f'(x)=5(12x^2+36)^4\cdot24x=120x(12x^2+36)^4$

Step3: Relate derivative to differential

The differential $df=f'(x)dx$

Answer:

$df = 120x(12x^2+36)^4dx$ (the fourth option)