Calculus Examples

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

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = e^{x}$ and $g (x) = \sin (x) + \cos (x)$ .

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

Step 2

By the Sum Rule, the derivative of $\sin (x) + \cos (x)$ with respect to $x$ is $\frac{d}{d x} [\sin (x)] + \frac{d}{d x} [\cos (x)]$ .

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Combine terms.

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Step 6.3.1

Reorder $\sin (x)$ and $e^{x}$ .

$e^{x} \cos (x) - 1 \cdot e^{x} \sin (x) + e^{x} \sin (x) + \cos (x) e^{x}$

Step 6.3.2

Rewrite $- 1 e^{x}$ as $- e^{x}$ .

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

Step 6.3.3

Add $- e^{x} \sin (x)$ and $e^{x} \sin (x)$ .

$e^{x} \cos (x) + 0 + \cos (x) e^{x}$

Step 6.3.4

Add $e^{x} \cos (x)$ and $0$ .

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

Step 6.3.5

Add $e^{x} \cos (x)$ and $\cos (x) e^{x}$ .

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Step 6.3.5.1

Reorder $\cos (x)$ and $e^{x}$ .

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

Step 6.3.5.2

Add $e^{x} \cos (x)$ and $e^{x} \cos (x)$ .

$2 e^{x} \cos (x)$