Topics
>
AP Calculus AB>
Derivatives>
Derivatives of Exponential Functions>
Derivatives of Exponential Functions 2Derivatives of Exponential Functions 2
Try VividMath Premium to unlock full access
Time limit: 0
Quiz summary
0 of 5 questions completed
Questions:
- 1
- 2
- 3
- 4
- 5
Information
–
You have already completed the quiz before. Hence you can not start it again.
Quiz is loading...
You must sign in or sign up to start the quiz.
You have to finish following quiz, to start this quiz:
Loading...
- 1
- 2
- 3
- 4
- 5
- Answered
- Review
-
Question 1 of 5
1. Question
Find the derivative`y=e^(2x)`Hint
Help VideoCorrect
Great Work!
Incorrect
Product Rule with Base “e”
$$\frac{d}{dx}(e^{\color{#D800AD}{f(x)}})=\color{#9a00c7}{f'(x)}\cdot e^{\color{#D800AD}{f(x)}}$$Substitute the components into the formula$$\frac{d}{dx}(e^{\color{#D800AD}{f(x)}})$$ `=` $$\color{#9a00c7}{f'(x)}\cdot e^{\color{#D800AD}{f(x)}}$$ `=` $$\color{#9a00c7}{f'(2x)}\cdot e^{\color{#D800AD}{2x}}$$ Substitute known values `=` $$\color{#9a00c7}{2}\cdot e^{\color{#D800AD}{2x}}$$ Differentiate `2x` `y’` `=` `2e^(2x)` `d/dx (e^(f(x)))=y’` `y’=2e^(2x)` -
Question 2 of 5
2. Question
Find the derivative`y=e^(4x)+5`Hint
Help VideoCorrect
Nice Job!
Incorrect
Product Rule with Base “e”
$$\frac{d}{dx}(e^{\color{#D800AD}{f(x)}})=\color{#9a00c7}{f'(x)}\cdot e^{\color{#D800AD}{f(x)}}$$Substitute the components into the formulaDifferentiating constants makes them `0`$$\frac{d}{dx}(e^{\color{#D800AD}{f(x)}})$$ `=` $$\color{#9a00c7}{f'(x)}\cdot e^{\color{#D800AD}{f(x)}}$$ `=` $$\color{#9a00c7}{f'(4x)}\cdot e^{\color{#D800AD}{4x}}+5$$ Substitute known values `=` $$\color{#9a00c7}{4}\cdot e^{4x}+0$$ Differentiate `4x` and `5` `y’` `=` `4e^(4x)` `d/dx (e^(f(x)))=y’` `y’=4e^(4x)` -
Question 3 of 5
3. Question
Find the derivative`y=e^(-3x)`Hint
Help VideoCorrect
Fantastic!
Incorrect
Product Rule with Base “e”
$$\frac{d}{dx}(e^{\color{#D800AD}{f(x)}})=\color{#9a00c7}{f'(x)}\cdot e^{\color{#D800AD}{f(x)}}$$Substitute the components into the formula$$\frac{d}{dx}(e^{\color{#D800AD}{f(x)}})$$ `=` $$\color{#9a00c7}{f'(x)}\cdot e^{\color{#D800AD}{f(x)}}$$ `=` $$\color{#9a00c7}{f'(-3x)}\cdot e^{\color{#D800AD}{-3x}}$$ Substitute known values `=` $$\color{#9a00c7}{-3}\cdot e^{-3x}$$ Differentiate `-3x` `y’` `=` `-3e^(-3x)` `d/dx (e^(f(x)))=y’` `y’=-3e^(-3x)` -
Question 4 of 5
4. Question
Find the derivative`y=2e^(-x/2)`Hint
Help VideoCorrect
Correct!
Incorrect
Product Rule with Base “e”
$$\frac{d}{dx}(e^{\color{#D800AD}{f(x)}})=\color{#9a00c7}{f'(x)}\cdot e^{\color{#D800AD}{f(x)}}$$Substitute the components into the formula$$\frac{d}{dx}(e^{\color{#D800AD}{f(x)}})$$ `=` $$\color{#9a00c7}{f'(x)}\cdot e^{\color{#D800AD}{f(x)}}$$ `=` $$\color{#9a00c7}{f'(-\frac{x}{2})}\cdot 2e^{\color{#D800AD}{-\frac{x}{2}}}$$ Substitute known values `=` $$\color{#9a00c7}{-\frac{1}{2}}\cdot 2e^{-\frac{x}{2}}$$ Differentiate `-x/2` `y’` `=` `-e^(-x/2)` `d/dx (e^(f(x)))=y’` `y’=-e^(-x/2)` -
Question 5 of 5
5. Question
Given that `y=e^(px)`, find `p``(d^2y)/(dx^2)-(dy)/(dx)-6y=0`- `p=` (3, -2) or (-2, 3)
Hint
Help VideoCorrect
Correct!
Incorrect
`y’=``(dy)/dx``y”=``(d^2y)/(dx^2)`First, find the first and second derivative of `y`First Derivative:`y` `=` `e^(px)` `y’` `=` `pe^(px)` Second Derivative:`y` `=` `e^(px)` `y”` `=` `p^2e^(px)` Substitute the components into the equation$$\color{#004ec4}{\frac{d^2y}{dx^2}}-\color{#e65021}{\frac{dy}{dx}}-6\color{#00880A}{y}$$ `=` `0` $$\color{#004ec4}{p^2e^{px}}-\color{#e65021}{pe^{px}}-6\color{#00880A}{e^{px}}$$ `=` `0` Substitute known values `(p^2e^(px)-pe^(px)-6e^(px))``divide e^(px)` `=` `0``divide e^(px)` Divide both sides by `e^(px)` `p^2-p-6` `=` `0` Evaluate `(p-3)(p+2)` `=` `0` Factorize Therefore, the value of `p` is `3` and `-2`.`p=3,-2`
Quizzes
- Power Rule 1
- Power Rule 2
- Power Rule 3
- Power Rule 4
- Chain Rule 1
- Chain Rule 2
- Product Rule
- Quotient Rule
- Derivatives of Exponential Functions 1
- Derivatives of Exponential Functions 2
- Derivatives of Exponential Functions 3
- Derivatives of Trigonometric Functions 1
- Derivatives of Trigonometric Functions 2
- Derivatives of Trigonometric Functions 3