Calculus Examples

$\frac{d}{d x} (u v)$

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 $f (u) = u$ and $g (u) = v$ .

$u \frac{d}{d u} [v] + v \frac{d}{d u} [u]$

Step 2

Differentiate.

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Step 2.1

Since $v$ is constant with respect to $u$ , the derivative of $v$ with respect to $u$ is $0$ .

$u \cdot 0 + v \frac{d}{d u} [u]$

Step 2.2

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

$u \cdot 0 + v \cdot 1$

Step 3

Combine terms.

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Step 3.1

Multiply $u$ by $0$ .

$0 + v \cdot 1$

Step 3.2

Multiply $v$ by $1$ .

$0 + v$

Step 3.3

Add $0$ and $v$ .

$v$