Find the slope of the tangent to the curve [tex]y=3+4 x^{2}-2 x^{3}[/tex] at the point where [tex]x=a\text{.}[/tex]
The derivative of a function can be determined using different rules and formulations. One of the important rules here is the power rule. Power rule is used to differentiate functions in the form of [tex]f(x) = x^n[/tex].
[tex]\; \dfrac{d}{dx} (x^n) = nx^{n-1}[/tex]
We are supposed to find the slope of the tangent to the curve [tex]y=3+4 x^{2}-2 x^{3}[/tex] at the point where [tex]x=a[/tex].
[tex]\implies \dfrac{d}{dx}(3+4x^2-2x^3)[/tex]
[tex]\implies \dfrac{d}{dx}(0+4(2x^{2-1})-2(3x^{3-1})[/tex]
[tex]\implies \dfrac{d}{dx}(0+4(2x)-2(3x^2)[/tex]
[tex]\implies \dfrac{d}{dx}(8x-6x^2[/tex]
Now plugging [tex]x=a[/tex] in the above equation, we obtain the following results:
[tex]\implies \boxed{8a -6a^2}[/tex]
Hence, this is our required solution for this question.
