Calculus Examples

$x^{2} y - 2 = 5 x + y$

Step 1

Find the first derivative.

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Step 1.1

By the Sum Rule, the derivative of $5 x + y$ with respect to $x$ is $\frac{d}{d x} [5 x] + \frac{d}{d x} [y]$ .

$\frac{d}{d x} [5 x] + \frac{d}{d x} [y]$

Step 1.2

Evaluate $\frac{d}{d x} [5 x]$ .

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Step 1.2.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x] + \frac{d}{d x} [y]$

Step 1.2.2

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

$5 \cdot 1 + \frac{d}{d x} [y]$

Step 1.2.3

Multiply $5$ by $1$ .

$5 + \frac{d}{d x} [y]$

Step 1.3

Differentiate using the Constant Rule.

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Step 1.3.1

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

$5 + 0$

Step 1.3.2

Add $5$ and $0$ .

$f' (x) = 5$

Step 2

Since $5$ is constant with respect to $x$ , the derivative of $5$ with respect to $x$ is $0$ .

$f'' (x) = 0$