Calculus Examples

$\sin (x^{2} y^{2})$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = \sin (x)$ and $g (x) = x^{2} y^{2}$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $x^{2} y^{2}$ .

$\frac{d}{d u} [\sin (u)] \frac{d}{d x} [x^{2} y^{2}]$

Step 1.2

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

$\cos (u) \frac{d}{d x} [x^{2} y^{2}]$

Step 1.3

Replace all occurrences of $u$ with $x^{2} y^{2}$ .

$\cos (x^{2} y^{2}) \frac{d}{d x} [x^{2} y^{2}]$

Step 2

Differentiate.

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Step 2.1

Since $y^{2}$ is constant with respect to $x$ , the derivative of $x^{2} y^{2}$ with respect to $x$ is $y^{2} \frac{d}{d x} [x^{2}]$ .

$\cos (x^{2} y^{2}) (y^{2} \frac{d}{d x} [x^{2}])$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

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

Step 2.3

Reorder.

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Step 2.3.1

Move $2$ to the left of $y^{2}$ .

$\cos (x^{2} y^{2}) (2 \cdot y^{2} x)$

Step 2.3.2

Reorder the factors of $\cos (x^{2} y^{2}) (2 y^{2} x)$ .

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