Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1 + u^{2}}$ .

$\frac{1}{1 + u^{2}} \frac{d}{d x} [x^{2} y]$

Step 1.3

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

$\frac{1}{1 + {(x^{2} y)}^{2}} \frac{d}{d x} [x^{2} y]$

Step 2

Differentiate.

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Step 2.1

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

$\frac{1}{1 + {(x^{2} y)}^{2}} (y \frac{d}{d x} [x^{2}])$

Step 2.2

Combine $y$ and $\frac{1}{1 + {(x^{2} y)}^{2}}$ .

$\frac{y}{1 + {(x^{2} y)}^{2}} \frac{d}{d x} [x^{2}]$

Step 2.3

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

$\frac{y}{1 + {(x^{2} y)}^{2}} (2 x)$

Step 2.4

Combine fractions.

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Step 2.4.1

Combine $2$ and $\frac{y}{1 + {(x^{2} y)}^{2}}$ .

$\frac{2 y}{1 + {(x^{2} y)}^{2}} x$

Step 2.4.2

Combine $\frac{2 y}{1 + {(x^{2} y)}^{2}}$ and $x$ .

$\frac{2 y x}{1 + {(x^{2} y)}^{2}}$

Step 3

Simplify.

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Step 3.1

Apply the product rule to $x^{2} y$ .

$\frac{2 y x}{1 + {(x^{2})}^{2} y^{2}}$

Step 3.2

Multiply the exponents in ${(x^{2})}^{2}$ .

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Step 3.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{2 y x}{1 + x^{2 \cdot 2} y^{2}}$

Step 3.2.2

Multiply $2$ by $2$ .

$\frac{2 y x}{1 + x^{4} y^{2}}$

Step 3.3

Reorder terms.

$\frac{2 x y}{x^{4} y^{2} + 1}$