Calculus Examples

$16 {(x^{2} + y^{2})}^{2}$

Step 1

Since $16$ is constant with respect to $x$ , the derivative of $16 {(x^{2} + y^{2})}^{2}$ with respect to $x$ is $16 \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]$ .

$16 \frac{d}{d x} [{(x^{2} + y^{2})}^{2}]$

Step 2

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

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Step 2.1

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

$16 (\frac{d}{d u} [u^{2}] \frac{d}{d x} [x^{2} + y^{2}])$

Step 2.2

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

$16 (2 u \frac{d}{d x} [x^{2} + y^{2}])$

Step 2.3

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

$16 (2 (x^{2} + y^{2}) \frac{d}{d x} [x^{2} + y^{2}])$

Step 3

Differentiate.

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Step 3.1

Multiply $2$ by $16$ .

$32 ((x^{2} + y^{2}) \frac{d}{d x} [x^{2} + y^{2}])$

Step 3.2

By the Sum Rule, the derivative of $x^{2} + y^{2}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}]$ .

$32 (x^{2} + y^{2}) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [y^{2}])$

Step 3.3

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

$32 (x^{2} + y^{2}) (2 x + \frac{d}{d x} [y^{2}])$

Step 3.4

Since $y^{2}$ is constant with respect to $x$ , the derivative of $y^{2}$ with respect to $x$ is $0$ .

$32 (x^{2} + y^{2}) (2 x + 0)$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $2 x$ and $0$ .

$32 (x^{2} + y^{2}) (2 x)$

Step 3.5.2

Multiply $2$ by $32$ .

$64 (x^{2} + y^{2}) x$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$(64 x^{2} + 64 y^{2}) x$

Step 4.2

Apply the distributive property.

$64 x^{2} x + 64 y^{2} x$

Step 4.3

Combine terms.

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Step 4.3.1

Raise $x$ to the power of $1$ .

$64 (x^{1} x^{2}) + 64 y^{2} x$

Step 4.3.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$64 x^{1 + 2} + 64 y^{2} x$

Step 4.3.3

Add $1$ and $2$ .

$64 x^{3} + 64 y^{2} x$