Calculus Examples

$3 x - 2 y = x^{2} - 2$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 x - 2 y) = \frac{d}{d x} (x^{2} - 2)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $3 x - 2 y$ with respect to $x$ is $\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2 y]$ .

$\frac{d}{d x} [3 x] + \frac{d}{d x} [- 2 y]$

Step 2.2

Evaluate $\frac{d}{d x} [3 x]$ .

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Step 2.2.1

Since $3$ is constant with respect to $x$ , the derivative of $3 x$ with respect to $x$ is $3 \frac{d}{d x} [x]$ .

$3 \frac{d}{d x} [x] + \frac{d}{d x} [- 2 y]$

Step 2.2.2

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

$3 \cdot 1 + \frac{d}{d x} [- 2 y]$

Step 2.2.3

Multiply $3$ by $1$ .

$3 + \frac{d}{d x} [- 2 y]$

Step 2.3

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

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Step 2.3.1

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

$3 - 2 \frac{d}{d x} [y]$

Step 2.3.2

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

$3 - 2 y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

$2 x + 0$

Step 3.4

Add $2 x$ and $0$ .

$2 x$

Step 4

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

$3 - 2 y' = 2 x$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $3$ from both sides of the equation.

$- 2 y' = 2 x - 3$

Step 5.2

Divide each term in $- 2 y' = 2 x - 3$ by $- 2$ and simplify.

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Step 5.2.1

Divide each term in $- 2 y' = 2 x - 3$ by $- 2$ .

$\frac{- 2 y'}{- 2} = \frac{2 x}{- 2} + \frac{- 3}{- 2}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $- 2$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{- 2 y'}{- 2} = \frac{2 x}{- 2} + \frac{- 3}{- 2}$

Step 5.2.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{2 x}{- 2} + \frac{- 3}{- 2}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Simplify each term.

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Step 5.2.3.1.1

Cancel the common factor of $2$ and $- 2$ .

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Step 5.2.3.1.1.1

Factor $2$ out of $2 x$ .

$y' = \frac{2 (x)}{- 2} + \frac{- 3}{- 2}$

Step 5.2.3.1.1.2

Move the negative one from the denominator of $\frac{x}{- 1}$ .

$y' = - 1 \cdot x + \frac{- 3}{- 2}$

Step 5.2.3.1.2

Rewrite $- 1 \cdot x$ as $- x$ .

$y' = - x + \frac{- 3}{- 2}$

Step 5.2.3.1.3

Dividing two negative values results in a positive value.

$y' = - x + \frac{3}{2}$

Step 6

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

$\frac{d y}{d x} = - x + \frac{3}{2}$