Calculus Examples

$y = u \cos (u)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d u} (y) = \frac{d}{d u} (u \cos (u))$

Step 2

The derivative of $y$ with respect to $u$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Differentiate using the Power Rule.

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Step 3.3.1

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

$u (- \sin (u)) + \cos (u) \cdot 1$

Step 3.3.2

Simplify the expression.

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Step 3.3.2.1

Multiply $\cos (u)$ by $1$ .

$u (- \sin (u)) + \cos (u)$

Step 3.3.2.2

Reorder terms.

$- u \sin (u) + \cos (u)$

Step 4

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

$y' = - u \sin (u) + \cos (u)$

Step 5

Replace $y'$ with $\frac{d y}{d u}$ .

$\frac{d y}{d u} = - u \sin (u) + \cos (u)$