Calculus Examples

$y'''' + y'''''''' = 6 e^{2 x}$ , $y = e^{2 x}$

Step 1

Find $y'$ .

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Step 1.1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (e^{2 x})$

Step 1.2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 1.3

Differentiate the right side of the equation.

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Step 1.3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 1.3.1.1

To apply the Chain Rule, set $u$ as $2 x$ .

$\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x]$

Step 1.3.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$e^{u} \frac{d}{d x} [2 x]$

Step 1.3.1.3

Replace all occurrences of $u$ with $2 x$ .

$e^{2 x} \frac{d}{d x} [2 x]$

Step 1.3.2

Differentiate.

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Step 1.3.2.1

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$e^{2 x} (2 \frac{d}{d x} [x])$

Step 1.3.2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$e^{2 x} (2 \cdot 1)$

Step 1.3.2.3

Simplify the expression.

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Step 1.3.2.3.1

Multiply $2$ by $1$ .

$e^{2 x} \cdot 2$

Step 1.3.2.3.2

Move $2$ to the left of $e^{2 x}$ .

$2 e^{2 x}$

Step 1.4

Reform the equation by setting the left side equal to the right side.

$y' = 2 e^{2 x}$

Step 2

Find $y''$ .

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Step 2.1

Set up the derivative.

$y'' = \frac{d}{d x} [2 e^{2 x}]$

Step 2.2

Since $2$ is constant with respect to $x$ , the derivative of $2 e^{2 x}$ with respect to $x$ is $2 \frac{d}{d x} [e^{2 x}]$ .

$y'' = 2 \frac{d}{d x} [e^{2 x}]$

Step 2.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 2.3.1

To apply the Chain Rule, set $u$ as $2 x$ .

$y'' = 2 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x])$

Step 2.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$y'' = 2 (e^{u} \frac{d}{d x} [2 x])$

Step 2.3.3

Replace all occurrences of $u$ with $2 x$ .

$y'' = 2 (e^{2 x} \frac{d}{d x} [2 x])$

Step 2.4

Remove parentheses.

$y'' = 2 e^{2 x} \frac{d}{d x} [2 x]$

Step 2.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$y'' = 2 e^{2 x} (2 \frac{d}{d x} [x])$

Step 2.6

Multiply $2$ by $2$ .

$y'' = 4 e^{2 x} \frac{d}{d x} [x]$

Step 2.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$y'' = 4 e^{2 x} \cdot 1$

Step 2.8

Multiply $4$ by $1$ .

$y'' = 4 e^{2 x}$

Step 3

Find $y'''$ .

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Step 3.1

Set up the derivative.

$y''' = \frac{d}{d x} [4 e^{2 x}]$

Step 3.2

Since $4$ is constant with respect to $x$ , the derivative of $4 e^{2 x}$ with respect to $x$ is $4 \frac{d}{d x} [e^{2 x}]$ .

$y''' = 4 \frac{d}{d x} [e^{2 x}]$

Step 3.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 3.3.1

To apply the Chain Rule, set $u$ as $2 x$ .

$y''' = 4 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x])$

Step 3.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$y''' = 4 (e^{u} \frac{d}{d x} [2 x])$

Step 3.3.3

Replace all occurrences of $u$ with $2 x$ .

$y''' = 4 (e^{2 x} \frac{d}{d x} [2 x])$

Step 3.4

Remove parentheses.

$y''' = 4 e^{2 x} \frac{d}{d x} [2 x]$

Step 3.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$y''' = 4 e^{2 x} (2 \frac{d}{d x} [x])$

Step 3.6

Multiply $2$ by $4$ .

$y''' = 8 e^{2 x} \frac{d}{d x} [x]$

Step 3.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$y''' = 8 e^{2 x} \cdot 1$

Step 3.8

Multiply $8$ by $1$ .

$y''' = 8 e^{2 x}$

Step 4

Find $y''''$ .

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Step 4.1

Set up the derivative.

$y'''' = \frac{d}{d x} [8 e^{2 x}]$

Step 4.2

Since $8$ is constant with respect to $x$ , the derivative of $8 e^{2 x}$ with respect to $x$ is $8 \frac{d}{d x} [e^{2 x}]$ .

$y'''' = 8 \frac{d}{d x} [e^{2 x}]$

Step 4.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 4.3.1

To apply the Chain Rule, set $u$ as $2 x$ .

$y'''' = 8 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x])$

Step 4.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$y'''' = 8 (e^{u} \frac{d}{d x} [2 x])$

Step 4.3.3

Replace all occurrences of $u$ with $2 x$ .

$y'''' = 8 (e^{2 x} \frac{d}{d x} [2 x])$

Step 4.4

Remove parentheses.

$y'''' = 8 e^{2 x} \frac{d}{d x} [2 x]$

Step 4.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$y'''' = 8 e^{2 x} (2 \frac{d}{d x} [x])$

Step 4.6

Multiply $2$ by $8$ .

$y'''' = 16 e^{2 x} \frac{d}{d x} [x]$

Step 4.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$y'''' = 16 e^{2 x} \cdot 1$

Step 4.8

Multiply $16$ by $1$ .

$y'''' = 16 e^{2 x}$

Step 5

Find $y'''''$ .

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Step 5.1

Set up the derivative.

$y''''' = \frac{d}{d x} [16 e^{2 x}]$

Step 5.2

Since $16$ is constant with respect to $x$ , the derivative of $16 e^{2 x}$ with respect to $x$ is $16 \frac{d}{d x} [e^{2 x}]$ .

$y''''' = 16 \frac{d}{d x} [e^{2 x}]$

Step 5.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 5.3.1

To apply the Chain Rule, set $u$ as $2 x$ .

$y''''' = 16 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x])$

Step 5.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$y''''' = 16 (e^{u} \frac{d}{d x} [2 x])$

Step 5.3.3

Replace all occurrences of $u$ with $2 x$ .

$y''''' = 16 (e^{2 x} \frac{d}{d x} [2 x])$

Step 5.4

Remove parentheses.

$y''''' = 16 e^{2 x} \frac{d}{d x} [2 x]$

Step 5.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$y''''' = 16 e^{2 x} (2 \frac{d}{d x} [x])$

Step 5.6

Multiply $2$ by $16$ .

$y''''' = 32 e^{2 x} \frac{d}{d x} [x]$

Step 5.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$y''''' = 32 e^{2 x} \cdot 1$

Step 5.8

Multiply $32$ by $1$ .

$y''''' = 32 e^{2 x}$

Step 6

Find $y''''''$ .

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Step 6.1

Set up the derivative.

$y'''''' = \frac{d}{d x} [32 e^{2 x}]$

Step 6.2

Since $32$ is constant with respect to $x$ , the derivative of $32 e^{2 x}$ with respect to $x$ is $32 \frac{d}{d x} [e^{2 x}]$ .

$y'''''' = 32 \frac{d}{d x} [e^{2 x}]$

Step 6.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 6.3.1

To apply the Chain Rule, set $u$ as $2 x$ .

$y'''''' = 32 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x])$

Step 6.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$y'''''' = 32 (e^{u} \frac{d}{d x} [2 x])$

Step 6.3.3

Replace all occurrences of $u$ with $2 x$ .

$y'''''' = 32 (e^{2 x} \frac{d}{d x} [2 x])$

Step 6.4

Remove parentheses.

$y'''''' = 32 e^{2 x} \frac{d}{d x} [2 x]$

Step 6.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$y'''''' = 32 e^{2 x} (2 \frac{d}{d x} [x])$

Step 6.6

Multiply $2$ by $32$ .

$y'''''' = 64 e^{2 x} \frac{d}{d x} [x]$

Step 6.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$y'''''' = 64 e^{2 x} \cdot 1$

Step 6.8

Multiply $64$ by $1$ .

$y'''''' = 64 e^{2 x}$

Step 7

Find $y'''''''$ .

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Step 7.1

Set up the derivative.

$y''''''' = \frac{d}{d x} [64 e^{2 x}]$

Step 7.2

Since $64$ is constant with respect to $x$ , the derivative of $64 e^{2 x}$ with respect to $x$ is $64 \frac{d}{d x} [e^{2 x}]$ .

$y''''''' = 64 \frac{d}{d x} [e^{2 x}]$

Step 7.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 7.3.1

To apply the Chain Rule, set $u$ as $2 x$ .

$y''''''' = 64 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x])$

Step 7.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$y''''''' = 64 (e^{u} \frac{d}{d x} [2 x])$

Step 7.3.3

Replace all occurrences of $u$ with $2 x$ .

$y''''''' = 64 (e^{2 x} \frac{d}{d x} [2 x])$

Step 7.4

Remove parentheses.

$y''''''' = 64 e^{2 x} \frac{d}{d x} [2 x]$

Step 7.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$y''''''' = 64 e^{2 x} (2 \frac{d}{d x} [x])$

Step 7.6

Multiply $2$ by $64$ .

$y''''''' = 128 e^{2 x} \frac{d}{d x} [x]$

Step 7.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$y''''''' = 128 e^{2 x} \cdot 1$

Step 7.8

Multiply $128$ by $1$ .

$y''''''' = 128 e^{2 x}$

Step 8

Find $y''''''''$ .

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Step 8.1

Set up the derivative.

$y'''''''' = \frac{d}{d x} [128 e^{2 x}]$

Step 8.2

Since $128$ is constant with respect to $x$ , the derivative of $128 e^{2 x}$ with respect to $x$ is $128 \frac{d}{d x} [e^{2 x}]$ .

$y'''''''' = 128 \frac{d}{d x} [e^{2 x}]$

Step 8.3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = 2 x$ .

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Step 8.3.1

To apply the Chain Rule, set $u$ as $2 x$ .

$y'''''''' = 128 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [2 x])$

Step 8.3.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u} [a^{u}]$ is $a^{u} \ln (a)$ where $a$ = $e$ .

$y'''''''' = 128 (e^{u} \frac{d}{d x} [2 x])$

Step 8.3.3

Replace all occurrences of $u$ with $2 x$ .

$y'''''''' = 128 (e^{2 x} \frac{d}{d x} [2 x])$

Step 8.4

Remove parentheses.

$y'''''''' = 128 e^{2 x} \frac{d}{d x} [2 x]$

Step 8.5

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$y'''''''' = 128 e^{2 x} (2 \frac{d}{d x} [x])$

Step 8.6

Multiply $2$ by $128$ .

$y'''''''' = 256 e^{2 x} \frac{d}{d x} [x]$

Step 8.7

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$y'''''''' = 256 e^{2 x} \cdot 1$

Step 8.8

Multiply $256$ by $1$ .

$y'''''''' = 256 e^{2 x}$

Step 9

Substitute into the given differential equation.

$16 e^{2 x} + 256 e^{2 x} = 6 e^{2 x}$

Step 10

Add $16 e^{2 x}$ and $256 e^{2 x}$ .

$272 e^{2 x} = 6 e^{2 x}$

Step 11

The given solution does not satisfy the given differential equation.

$y = e^{2 x}$ is not a solution to $y'''' + y'''''''' = 6 e^{2 x}$