We are given the function:
[tex]y=\frac{3}{(2x)^{-2}}[/tex]
To rewrite this, we need to remember that negative exponents mean the reciprocal of the expression. So we get:
[tex]\begin{gathered} y=\frac{3}{(2x)^{-2}} \\ \\ y=3(2x)^2 \\ \\ y=3(4x^2) \\ \\ y=12x^2 \end{gathered}[/tex]
To differentiate, we bring down the exponent as a multiplier, then reduce the original exponent by 1.
[tex]\begin{gathered} y^{\prime}=12(2)x^{2-1} \\ \\ y^{\prime}=24x \end{gathered}[/tex]
The derivative of f(x) = x^a is f'(x) = ax^(a-1).
[tex]\begin{gathered} f(x)=x^a \\ f^{\prime}(x)=ax^{a-1} \end{gathered}[/tex]
So if we are looking for f'(x) when f(x) = x^2,
[tex]\begin{gathered} f(x)=x^2 \\ f^{\prime}(x)=2x^{2-1} \\ f^{\prime}(x)=2x \end{gathered}[/tex]
Applying this to our function f(x) = 12x^2, we get:
[tex]\begin{gathered} f(x)=12x^2 \\ f^{\prime}(x)=12(2)x^{2-1} \\ f^{\prime}(x)=24x^1 \\ f^{\prime}(x)=24x \end{gathered}[/tex]
