Calculus Examples

$y - x^{2} + 2 x = 0$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y - x^{2} + 2 x) = \frac{d}{d x} (0)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

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

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

Step 2.2

Rewrite $\frac{d}{d x} [y]$ as $y'$ .

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

Step 2.3

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

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Step 2.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

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

Step 2.3.2

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

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

Step 2.3.3

Multiply $2$ by $- 1$ .

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

Step 2.4

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

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Step 2.4.1

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

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

Step 2.4.2

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

$y' - 2 x + 2 \cdot 1$

Step 2.4.3

Multiply $2$ by $1$ .

$y' - 2 x + 2$

Step 2.5

Reorder terms.

$- 2 x + y' + 2$

Step 3

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

$0$

Step 4

Reform the equation by setting the left side equal to the right side.

$- 2 x + y' + 2 = 0$

Step 5

Move all terms not containing $y'$ to the right side of the equation.

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Step 5.1

Add $2 x$ to both sides of the equation.

$y' + 2 = 2 x$

Step 5.2

Subtract $2$ from both sides of the equation.

$y' = 2 x - 2$

Step 6

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 2 x - 2$