Calculus Examples

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

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

Step 1

Differentiate both sides of the equation.

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

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

Step 2

The derivative of

y

$y$ with respect to

x

$x$ is

y'

$y'$ .

y'

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Product Rule which states that

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

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

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

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

f (x) = e^{x}

$f (x) = e^{x}$ and

g (x) = \sin (x)

$g (x) = \sin (x)$ .

e^{x} \frac{d}{d x} [\sin (x)] + \sin (x) \frac{d}{d x} [e^{x}]

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

Step 3.2

The derivative of

\sin (x)

$\sin (x)$ with respect to

x

$x$ is

\cos (x)

$\cos (x)$ .

e^{x} \cos (x) + \sin (x) \frac{d}{d x} [e^{x}]

$e^{x} \cos (x) + \sin (x) \frac{d}{d x} [e^{x}]$

Step 3.3

Differentiate using the Exponential Rule which states that

\frac{d}{d x} [a^{x}]

$\frac{d}{d x} [a^{x}]$ is

a^{x} \ln (a)

$a^{x} \ln (a)$ where

a

$a$ =

e

$e$ .

e^{x} \cos (x) + \sin (x) e^{x}

$e^{x} \cos (x) + \sin (x) e^{x}$

Step 3.4

Reorder terms.

e^{x} \cos (x) + e^{x} \sin (x)

$e^{x} \cos (x) + e^{x} \sin (x)$

e^{x} \cos (x) + e^{x} \sin (x)

$e^{x} \cos (x) + e^{x} \sin (x)$

Step 4

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

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

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

Step 5

Replace

y'

$y'$ with

\frac{d y}{d x}

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

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

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

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

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

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