Calculus Examples

$(2 x + y) \cdot 3 = x$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} ((2 x + y) \cdot 3) = \frac{d}{d x} (x)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

Move $3$ to the left of $2 x + y$ .

$\frac{d}{d x} [3 \cdot (2 x + y)]$

Step 2.2

Since $3$ is constant with respect to $x$ , the derivative of $3 (2 x + y)$ with respect to $x$ is $3 \frac{d}{d x} [2 x + y]$ .

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Multiply $2$ by $1$ .

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

Step 2.7

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

$3 (2 + y')$

Step 2.8

Simplify.

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Step 2.8.1

Apply the distributive property.

$3 \cdot 2 + 3 y'$

Step 2.8.2

Multiply $3$ by $2$ .

$6 + 3 y'$

Step 2.8.3

Reorder terms.

$3 y' + 6$

Step 3

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

$1$

Step 4

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

$3 y' + 6 = 1$

Step 5

Solve for $y'$ .

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Step 5.1

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

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Step 5.1.1

Subtract $6$ from both sides of the equation.

$3 y' = 1 - 6$

Step 5.1.2

Subtract $6$ from $1$ .

$3 y' = - 5$

Step 5.2

Divide each term in $3 y' = - 5$ by $3$ and simplify.

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Step 5.2.1

Divide each term in $3 y' = - 5$ by $3$ .

$\frac{3 y'}{3} = \frac{- 5}{3}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $3$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{3 y'}{3} = \frac{- 5}{3}$

Step 5.2.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 5}{3}$

Step 5.2.3

Simplify the right side.

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Step 5.2.3.1

Move the negative in front of the fraction.

$y' = - \frac{5}{3}$

Step 6

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

$\frac{d y}{d x} = - \frac{5}{3}$