Question

question 2
find the derivative of $f(x)=(2x^3+3x-12)^6$.
$f(x)=6(2x^3+3x-12)^5$
$f(x)=6(6x^2+3)^5$
$f(x)=6(6x^2+3)(2x^3+3x-12)^5$
$f(x)=36x^{17}+18x^5$

Explanation:

Step1: Identify outer/inner functions

Let $u = 2x^3 + 3x - 12$, so $f(u) = u^6$.

Step2: Apply chain rule

Chain rule: $f'(x) = f'(u) \cdot u'(x)$.
First, $f'(u) = 6u^5 = 6(2x^3 + 3x - 12)^5$.
Then, $u'(x) = \frac{d}{dx}(2x^3 + 3x - 12) = 6x^2 + 3$.

Step3: Multiply the derivatives

$f'(x) = 6(2x^3 + 3x - 12)^5 \cdot (6x^2 + 3)$

Answer:

$f'(x)=6(6x^{2}+3)(2x^{3}+3x-12)^{5}$