Derivatives of Exponential Functions 3

Question 1 of 5

Find the derivative

$y=4-3e^(-x)$

1. $y'=-3e^(-x)$
2. $y'=-3e^(-x/2)$
3. $y'=e^(-3x)$
4. $y'=-3e^(-3x)$

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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

Differentiating constants makes them

$0$

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

	$=$	$4-\color{#9a00c7}{f'(-x)}\cdot 3e^{\color{#D800AD}{-x}}$	Substitute known values
	$=$	$0-(\color{#9a00c7}{-1}\cdot 3e^{-x})$	Differentiate $-x$ and $4$

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

$y’=-3e^(-x)$

Question 2 of 5

Find the derivative

$y=6e^(x/2)-3e^(-2x)$

1. $y'=3e^(x/2)+6e^(2x)$
2. $y'=6e^(-2x)$
3. $y'=3e^(x/2)$
4. $y'=3e^(x/2)+6e^(-2x)$

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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

First Term:

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

	$=$	$\left(\color{#9a00c7}{f’\left(\frac{x}{2}\right)}\cdot 6e^{\color{#D800AD}{\frac{x}{2}}}\right)-3e^{-2x}$	Substitute known values

	$=$	$\left(\color{#9a00c7}{\frac{1}{2}}\cdot 6e^{\frac{x}{2}}\right)-3e^{-2x}$	Diferrentiate $x/2$

	$=$	$3e^(x/2)-3e^(-2x)$

Second Term:

	$=$	$3e^\frac{x}{2}-\left(\color{#9a00c7}{f'(-2x)}\cdot 3e^{\color{#D800AD}{-2x}}\right)$	Substitute known values
	$=$	$3e^\frac{x}{2}-\left(\color{#9a00c7}{-2}\cdot 3e^{-2x}\right)$	Diferrentiate $-2x$

$y’$	$=$	$3e^(x/2)+6e^(-2x)$	$d/dx (e^(f(x)))=y’$

$y’=3e^(x/2)+6e^(-2x)$

Question 3 of 5

Find the derivative

$y=(e^(2x)+e^(-x))/2$

1. $y'=2e^x$
2. $y'=e^(2x)-(e^(-x))/2$
3. $y'=e^(2x)+(e^(-x))/2$
4. $y'=e^(2x)$

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Product Rule with Base “e”

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

Separate the two terms by giving each a denominator of

$2$

$(e^(2x)+e^(-x))/2$

$=$

$(e^(2x))/2+(e^(-x))/2$

Substitute the components into the formula

First Term:

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

	$=$	$\left(\color{#9a00c7}{f'(2x)}\cdot \frac{e^{\color{#D800AD}{2x}}}{2}\right)+\frac{e^{-x}}{2}$	Substitute known values

	$=$	$\left(\color{#9a00c7}{2}\cdot \frac{e^{2x}}{2}\right)+\frac{e^{-x}}{2}$	Diferrentiate $2x$

	$=$	$e^(2x)+(e^(-x))/2$	$2/2=1$

Second Term:

	$=$	$e^{2x}+\left(\color{#9a00c7}{f'(-x)}\cdot \frac{e^{\color{#D800AD}{-x}}}{2}\right)$	Substitute known values

	$=$	$e^{2x}+\left(\color{#9a00c7}{-1}\cdot \frac{e^{-x}}{2}\right)$	Differentiate $-x$

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

$y’=e^(2x)-(e^(-x))/2$

Question 4 of 5

Find the derivative

$y=e^(3-2x^2)$

1. $y'=-4xe^(3-2x^2)$
2. $y'=-4xe^(2x^2)$
3. $y'=e^(-3-2x^2)$
4. $y'=xe^(3-2x^2)$

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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

Differentiating constants makes them

$0$

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

	$=$	$\color{#9a00c7}{f'(3-2x^{2})}\cdot e^{\color{#D800AD}{3-2x^{2}}}$	Substitute known values
	$=$	$\color{#9a00c7}{-4x}\cdot e^{3-2x^{2}}$	Diferrentiate $3-2x^2$

$y’$	$=$	$-4xe^(3-2x^2)$	$d/dx (e^(f(x)))=y’$

$y’=-4xe^(3-2x^2)$

Question 5 of 5

Find the derivative

$y=e^(-0.03x)$

1. $y'=-0.03e^(-0.03x)$
2. $y'=0.03$
3. $y'=-0.03e^x$
4. $y'=0.03e^(-0.03x)$

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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

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

	$=$	$\color{#9a00c7}{f'(-0.03x)}\cdot e^{\color{#D800AD}{-0.03x}}$	Substitute known values
	$=$	$\color{#9a00c7}{-0.03}\cdot e^{-0.03x}$	Diferrentiate $-0.03x$

$y’$	$=$	$-0.03e^(-0.03x)$	$d/dx (e^(f(x)))=y’$

$y’=-0.03e^(-0.03x)$

Derivatives of Exponential Functions 3

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1. Question

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Product Rule with Base “e”

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Product Rule with Base “e”

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Product Rule with Base “e”

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Product Rule with Base “e”

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Product Rule with Base “e”

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