Question

Pre - Calculus evaluate exponential derivative at a point !

Answer 1

`\displaystyle`
`\displaystyle f'(1)=-(9)/(e^3)`

Explanation:

Use Quotient Rule to find f'(x)

`\displaystyle f(x)=(3x^2+2)/(e^(3x))\\\\f'(x)=(e^(3x)(6x)-(3x^2+2)(3e^(3x)))/((e^(3x))^2)\\\\f'(x)=(6xe^(3x)-(9x^2+6)(e^(3x)))/(e^(6x))\\\\f'(x)=(6x-(9x^2+6))/(e^(3x))\\\\f'(x)=(-9x^2+6x-6)/(e^(3x))`

Find f'(1) using f'(x)

`\displaystyle f'(1)=(-9(1)^2+6(1)-6)/(e^(3(1)))\\\\f'(1)=(-9+6-6)/(e^3)\\\\f'(1)=(-9)/(e^3)`

Answer 2

`f'(1)=-(9)/(e^(3))`

Explanation:

Given rational function:

`f(x)=(3x^2+2)/(e^(3x))`

To find the value of f'(1), we first need to differentiate the rational function to find f'(x). To do this, we can use the quotient rule.

`\boxed{\begin{minipage}{5.5 cm}\underline{Quotient Rule for Differentiation}\\\\If $f(x)=(g(x))/(h(x))$ then:\\\\\\$f'(x)=(h(x) g'(x)-g(x)h'(x))/((h(x))^2)$\\\end{minipage}}`

`\textsf{Let}\;g(x)=3x^2+2 \implies g'(x)=6x`

`\textsf{Let}\;h(x)=e^(3x) \implies h'(x)=3e^(3x)`

Therefore:

`f'(x)=(e^(3x) \cdot 6x -(3x^2+2) \cdot 3e^(3x))/(\left(e^(3x)\right)^2)`

`f'(x)=(6x -(3x^2+2) \cdot 3)/(e^(3x))`

`f'(x)=(6x -9x^2-6)/(e^(3x))`

To find f'(1), substitute x = 1 into f'(x):

`f'(1)=(6(1) -9(1)^2-6)/(e^(3(1)))`

`f'(1)=(6 -9-6)/(e^(3))`

`f'(1)=-(9)/(e^(3))`