Calculus Examples

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

Differentiate both sides of the equation.

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

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

$\cos (x)$

Differentiate the right side of the equation.

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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 $g (x) = y$ .

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

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

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

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

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

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

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

$e^{y} y'$

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

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

Solve for $y'$ .

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

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

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

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

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

Simplify the left side.

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Cancel the common factor of $e^{y}$ .

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Cancel the common factor.

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

Divide $y'$ by $1$ .

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

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

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