Calculus Examples

sin (x) = e^{y}

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

Differentiate both sides of the equation.

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

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

The derivative of

sin (x)

$\sin (x)$ with respect to

x

$x$ is

cos (x)

$\cos (x)$ .

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))]

$\frac{d}{d x} [f (g (x))]$ 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

$\frac{d y}{d x}$ .

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