Calculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

$2 x + 0 + \frac{d}{d x} [z^{2}]$

Step 4

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

$2 x + 0 + 0$

Step 5

Combine terms.

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Step 5.1

Add $2 x$ and $0$ .

$2 x + 0$

Step 5.2

Add $2 x$ and $0$ .

$2 x$