Calculus Examples

$y = e^{u} (\cos (u) + c u)$

Step 1

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

$e^{u} \frac{d}{d u} [\cos (u) + c u] + (\cos (u) + c u) \frac{d}{d u} [e^{u}]$

Step 2

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

$e^{u} (\frac{d}{d u} [\cos (u)] + \frac{d}{d u} [c u]) + (\cos (u) + c u) \frac{d}{d u} [e^{u}]$

Step 3

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

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

Step 4

Differentiate.

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Step 4.1

Since $c$ is constant with respect to $u$ , the derivative of $c u$ with respect to $u$ is $c \frac{d}{d u} [u]$ .

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

Step 4.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 1$ .

$e^{u} (- \sin (u) + c \cdot 1) + (\cos (u) + c u) \frac{d}{d u} [e^{u}]$

Step 4.3

Multiply $c$ by $1$ .

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

Step 5

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

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

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

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

Step 6.2

Apply the distributive property.

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

Step 6.3

Reorder terms.

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