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Pre - Calculus evaluate exponential derivative at a point !

Pre - Calculus evaluate exponential derivative at a point !-example-1

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7 votes

Answer:


\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)

User Teudimundo
by
7.6k points
4 votes

Answer:


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))

User Pmgarvey
by
7.7k points

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