Calculus Examples

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

Step 1

Differentiate both sides of the equation.

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

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.2

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

$\cos (x) x'$

Step 3

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

$e^{y}$

Step 4

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

$\cos (x) x' = e^{y}$

Step 5

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

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Step 5.1

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

$\frac{\cos (x) x'}{\cos (x)} = \frac{e^{y}}{\cos (x)}$

Step 5.2

Simplify the left side.

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Step 5.2.1

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

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Step 5.2.1.1

Cancel the common factor.

$\frac{\cos (x) x'}{\cos (x)} = \frac{e^{y}}{\cos (x)}$

Step 5.2.1.2

Divide $x'$ by $1$ .

$x' = \frac{e^{y}}{\cos (x)}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Separate fractions.

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

Step 5.3.2

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

$x' = \frac{e^{y}}{1} \sec (x)$

Step 5.3.3

Divide $e^{y}$ by $1$ .

$x' = e^{y} \sec (x)$

Step 6

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

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