find $\frac{dy}{dx}$ d) $f(x)=(2 - 3x^{4})^{2}$

Explanation:

Step1: Identify the outer - inner functions

Let $u = 2 - 3x^{4}$, then $y = u^{2}$.

Step2: Differentiate the outer function

The derivative of $y$ with respect to $u$ is $\frac{dy}{du}=2u$.

Step3: Differentiate the inner function

The derivative of $u$ with respect to $x$ is $\frac{du}{dx}=-12x^{3}$.

Step4: Apply the chain - rule

By the chain - rule $\frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}$. Substitute $u = 2 - 3x^{4}$, $\frac{dy}{du}=2u$ and $\frac{du}{dx}=-12x^{3}$ into the chain - rule formula. $\frac{dy}{dx}=2(2 - 3x^{4})\cdot(-12x^{3})=-24x^{3}(2 - 3x^{4})$.

Answer:

$-24x^{3}(2 - 3x^{4})$