Question

9. (a) Find the slope of the tangent to the curve
`y=3+4 x^(2)-2 x^(3)` at the point where
`x=a\text{.}`

Answer 1

Question:

Find the slope of the tangent to the curve
`y=3+4 x^(2)-2 x^(3)` at the point where
`x=a\text{.}`

Derivatives:

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
`f(x) = x^n`.

`\; (d)/(dx) (x^n) = nx^(n-1)`

Step-by-step explanation:

We are supposed to find the slope of the tangent to the curve
`y=3+4 x^(2)-2 x^(3)` at the point where
`x=a`.

`\implies (d)/(dx)(3+4x^2-2x^3)`

`\implies (d)/(dx)(0+4(2x^(2-1))-2(3x^(3-1))`

`\implies (d)/(dx)(0+4(2x)-2(3x^2)`

`\implies (d)/(dx)(8x-6x^2`

Now plugging
`x=a` in the above equation, we obtain the following results:

`\implies \boxed{8a -6a^2}`

Hence, this is our required solution for this question.