Derivatives of Exponential Functions 2

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Question 1 of 5

Find the derivative

`y=e^(2x)`

`y'=2e^(2x)`
`y'=e^x`
`y'=e^(2x)`
`y'=2e^x`

Question 2 of 5

Find the derivative

`y=e^(4x)+5`

`y'=4e^(4x)`
`y'=2e^x`
`y'=4e^(2x)`
`y'=4e^x`

Question 3 of 5

Find the derivative

`y=e^(-3x)`

`y'=-3e^(-3x)`
`y'=e^(-3x)`
`y'=-e^(3x)`
`y'=3e^(-3x)`

Question 4 of 5

Find the derivative

`y=2e^(-x/2)`

`y'=-e^(-x/2)`
`y'=-e^(x/2)`
`y'=e^(-x/2)`
`y'=e^(x/2)`

Question 5 of 5

Given that `y=e^(px)`, find `p`

`(d^2y)/(dx^2)-(dy)/(dx)-6y=0`

`p=` (3, -2) or (-2, 3)

$$\frac{d}{dx}(e^{\color{#D800AD}{f(x)}})$$	`=`	$$\color{#9a00c7}{f'(x)}\cdot e^{\color{#D800AD}{f(x)}}$$

	`=`	$$\color{#9a00c7}{f'(2x)}\cdot e^{\color{#D800AD}{2x}}$$	Substitute known values

	`=`	$$\color{#9a00c7}{2}\cdot e^{\color{#D800AD}{2x}}$$	Differentiate `2x`

`y’`	`=`	`2e^(2x)`	`d/dx (e^(f(x)))=y’`

$$\frac{d}{dx}(e^{\color{#D800AD}{f(x)}})$$	`=`	$$\color{#9a00c7}{f'(x)}\cdot e^{\color{#D800AD}{f(x)}}$$

	`=`	$$\color{#9a00c7}{f'(4x)}\cdot e^{\color{#D800AD}{4x}}+5$$	Substitute known values
	`=`	$$\color{#9a00c7}{4}\cdot e^{4x}+0$$	Differentiate `4x` and `5`

`y’`	`=`	`4e^(4x)`	`d/dx (e^(f(x)))=y’`

$$\frac{d}{dx}(e^{\color{#D800AD}{f(x)}})$$	`=`	$$\color{#9a00c7}{f'(x)}\cdot e^{\color{#D800AD}{f(x)}}$$

	`=`	$$\color{#9a00c7}{f'(-3x)}\cdot e^{\color{#D800AD}{-3x}}$$	Substitute known values
	`=`	$$\color{#9a00c7}{-3}\cdot e^{-3x}$$	Differentiate `-3x`

`y’`	`=`	`-3e^(-3x)`	`d/dx (e^(f(x)))=y’`

$$\frac{d}{dx}(e^{\color{#D800AD}{f(x)}})$$	`=`	$$\color{#9a00c7}{f'(x)}\cdot e^{\color{#D800AD}{f(x)}}$$

	`=`	$$\color{#9a00c7}{f'(-\frac{x}{2})}\cdot 2e^{\color{#D800AD}{-\frac{x}{2}}}$$	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^(px)`
`y’`	`=`	`pe^(px)`

Derivatives of Exponential Functions 2

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Quiz summary

Information

1. Question

Hint

Great Work!

Product Rule with Base “e”

2. Question

Hint

Nice Job!

Product Rule with Base “e”

3. Question

Hint

Fantastic!

Product Rule with Base “e”

4. Question

Hint

Correct!

Product Rule with Base “e”

5. Question

Hint

Correct!

Quizzes

$$\color{#004ec4}{\frac{d^2y}{dx^2}}-\color{#e65021}{\frac{dy}{dx}}-6\color{#00880A}{y}$$	`=`	`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