Calculus Examples

$y + 2 x - \sqrt{y^{4}}$

Step 1

Rewrite $y^{4}$ as ${(y^{2})}^{2}$ .

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

Step 2

Pull terms out from under the radical, assuming positive real numbers.

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

Step 3

Differentiate.

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Step 3.1

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

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

Step 3.2

Since $y$ is constant with respect to $x$ , the derivative of $y$ with respect to $x$ is $0$ .

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

Step 4

Evaluate $\frac{d}{d x} [2 x]$ .

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Step 4.1

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

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

Step 4.2

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

$0 + 2 \cdot 1 + \frac{d}{d x} [- y^{2}]$

Step 4.3

Multiply $2$ by $1$ .

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

Step 5

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

$0 + 2 + 0$

Step 6

Combine terms.

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Step 6.1

Add $0$ and $2$ .

$2 + 0$

Step 6.2

Add $2$ and $0$ .

$2$