Calculus Examples

$e^{2 x} = \sin (x + 8 y)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (e^{2 x}) = \frac{d}{d x} (\sin (x + 8 y))$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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 2.1.1

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

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

Step 2.1.2

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 2.1.3

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

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

Step 2.2

Differentiate.

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Step 2.2.1

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 2.2.2

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 2.2.3

Simplify the expression.

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Step 2.2.3.1

Multiply $2$ by $1$ .

$e^{2 x} \cdot 2$

Step 2.2.3.2

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

$2 e^{2 x}$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = x + 8 y$ .

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Step 3.1.1

To apply the Chain Rule, set $u$ as $x + 8 y$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [x + 8 y]$

Step 3.1.2

The derivative of $\sin (u)$ with respect to $u$ is $\cos (u)$ .

$\cos (u) \frac{d}{d x} [x + 8 y]$

Step 3.1.3

Replace all occurrences of $u$ with $x + 8 y$ .

$\cos (x + 8 y) \frac{d}{d x} [x + 8 y]$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $x + 8 y$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [8 y]$ .

$\cos (x + 8 y) (\frac{d}{d x} [x] + \frac{d}{d x} [8 y])$

Step 3.2.2

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

$\cos (x + 8 y) (1 + \frac{d}{d x} [8 y])$

Step 3.2.3

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

$\cos (x + 8 y) (1 + 8 \frac{d}{d x} [y])$

Step 3.3

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

$\cos (x + 8 y) (1 + 8 y')$

Step 4

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

$2 e^{2 x} = \cos (x + 8 y) (1 + 8 y')$

Step 5

Solve for $y'$ .

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Step 5.1

Rewrite the equation as $\cos (x + 8 y) (1 + 8 y') = 2 e^{2 x}$ .

$\cos (x + 8 y) (1 + 8 y') = 2 e^{2 x}$

Step 5.2

Divide each term in $\cos (x + 8 y) (1 + 8 y') = 2 e^{2 x}$ by $\cos (x + 8 y)$ and simplify.

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Step 5.2.1

Divide each term in $\cos (x + 8 y) (1 + 8 y') = 2 e^{2 x}$ by $\cos (x + 8 y)$ .

$\frac{\cos (x + 8 y) (1 + 8 y')}{\cos (x + 8 y)} = \frac{2 e^{2 x}}{\cos (x + 8 y)}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $\cos (x + 8 y)$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{\cos (x + 8 y) (1 + 8 y')}{\cos (x + 8 y)} = \frac{2 e^{2 x}}{\cos (x + 8 y)}$

Step 5.2.2.1.2

Divide $1 + 8 y'$ by $1$ .

$1 + 8 y' = \frac{2 e^{2 x}}{\cos (x + 8 y)}$

Step 5.3

Subtract $1$ from both sides of the equation.

$8 y' = \frac{2 e^{2 x}}{\cos (x + 8 y)} - 1$

Step 5.4

Divide each term in $8 y' = \frac{2 e^{2 x}}{\cos (x + 8 y)} - 1$ by $8$ and simplify.

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Step 5.4.1

Divide each term in $8 y' = \frac{2 e^{2 x}}{\cos (x + 8 y)} - 1$ by $8$ .

$\frac{8 y'}{8} = \frac{\frac{2 e^{2 x}}{\cos (x + 8 y)}}{8} + \frac{- 1}{8}$

Step 5.4.2

Simplify the left side.

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Step 5.4.2.1

Cancel the common factor of $8$ .

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Step 5.4.2.1.1

Cancel the common factor.

$\frac{8 y'}{8} = \frac{\frac{2 e^{2 x}}{\cos (x + 8 y)}}{8} + \frac{- 1}{8}$

Step 5.4.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{\frac{2 e^{2 x}}{\cos (x + 8 y)}}{8} + \frac{- 1}{8}$

Step 5.4.3

Simplify the right side.

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Step 5.4.3.1

Simplify each term.

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Step 5.4.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$y' = \frac{2 e^{2 x}}{\cos (x + 8 y)} \cdot \frac{1}{8} + \frac{- 1}{8}$

Step 5.4.3.1.2

Combine.

$y' = \frac{2 e^{2 x} \cdot 1}{\cos (x + 8 y) \cdot 8} + \frac{- 1}{8}$

Step 5.4.3.1.3

Cancel the common factor of $2$ and $8$ .

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Step 5.4.3.1.3.1

Factor $2$ out of $2 e^{2 x} \cdot 1$ .

$y' = \frac{2 (e^{2 x} \cdot 1)}{\cos (x + 8 y) \cdot 8} + \frac{- 1}{8}$

Step 5.4.3.1.3.2

Cancel the common factors.

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Step 5.4.3.1.3.2.1

Factor $2$ out of $\cos (x + 8 y) \cdot 8$ .

$y' = \frac{2 (e^{2 x} \cdot 1)}{2 (\cos (x + 8 y) \cdot 4)} + \frac{- 1}{8}$

Step 5.4.3.1.3.2.2

Cancel the common factor.

$y' = \frac{2 (e^{2 x} \cdot 1)}{2 (\cos (x + 8 y) \cdot 4)} + \frac{- 1}{8}$

Step 5.4.3.1.3.2.3

Rewrite the expression.

$y' = \frac{e^{2 x} \cdot 1}{\cos (x + 8 y) \cdot 4} + \frac{- 1}{8}$

Step 5.4.3.1.4

Multiply $e^{2 x}$ by $1$ .

$y' = \frac{e^{2 x}}{\cos (x + 8 y) \cdot 4} + \frac{- 1}{8}$

Step 5.4.3.1.5

Separate fractions.

$y' = \frac{e^{2 x}}{4} \cdot \frac{1}{\cos (x + 8 y)} + \frac{- 1}{8}$

Step 5.4.3.1.6

Convert from $\frac{1}{\cos (x + 8 y)}$ to $\sec (x + 8 y)$ .

$y' = \frac{e^{2 x}}{4} \sec (x + 8 y) + \frac{- 1}{8}$

Step 5.4.3.1.7

Combine $\frac{e^{2 x}}{4}$ and $\sec (x + 8 y)$ .

$y' = \frac{e^{2 x} \sec (x + 8 y)}{4} + \frac{- 1}{8}$

Step 5.4.3.1.8

Move the negative in front of the fraction.

$y' = \frac{e^{2 x} \sec (x + 8 y)}{4} - \frac{1}{8}$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = \frac{e^{2 x} \sec (x + 8 y)}{4} - \frac{1}{8}$