Calculus Examples

$7 y - 2 x^{2} = x - 9$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (7 y - 2 x^{2}) = \frac{d}{d x} (x - 9)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $7 y - 2 x^{2}$ with respect to $x$ is $\frac{d}{d x} [7 y] + \frac{d}{d x} [- 2 x^{2}]$ .

$\frac{d}{d x} [7 y] + \frac{d}{d x} [- 2 x^{2}]$

Step 2.2

Evaluate $\frac{d}{d x} [7 y]$ .

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Step 2.2.1

Since $7$ is constant with respect to $x$ , the derivative of $7 y$ with respect to $x$ is $7 \frac{d}{d x} [y]$ .

$7 \frac{d}{d x} [y] + \frac{d}{d x} [- 2 x^{2}]$

Step 2.2.2

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

$7 y' + \frac{d}{d x} [- 2 x^{2}]$

Step 2.3

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

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Step 2.3.1

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

$7 y' - 2 \frac{d}{d x} [x^{2}]$

Step 2.3.2

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

$7 y' - 2 (2 x)$

Step 2.3.3

Multiply $2$ by $- 2$ .

$7 y' - 4 x$

Step 2.4

Reorder terms.

$- 4 x + 7 y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

By the Sum Rule, the derivative of $x - 9$ with respect to $x$ is $\frac{d}{d x} [x] + \frac{d}{d x} [- 9]$ .

$\frac{d}{d x} [x] + \frac{d}{d x} [- 9]$

Step 3.2

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

$1 + \frac{d}{d x} [- 9]$

Step 3.3

Since $- 9$ is constant with respect to $x$ , the derivative of $- 9$ with respect to $x$ is $0$ .

$1 + 0$

Step 3.4

Add $1$ and $0$ .

$1$

Step 4

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

$- 4 x + 7 y' = 1$

Step 5

Solve for $y'$ .

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Step 5.1

Add $4 x$ to both sides of the equation.

$7 y' = 1 + 4 x$

Step 5.2

Divide each term in $7 y' = 1 + 4 x$ by $7$ and simplify.

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Step 5.2.1

Divide each term in $7 y' = 1 + 4 x$ by $7$ .

$\frac{7 y'}{7} = \frac{1}{7} + \frac{4 x}{7}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $7$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{7 y'}{7} = \frac{1}{7} + \frac{4 x}{7}$

Step 5.2.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{1}{7} + \frac{4 x}{7}$

Step 6

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

$\frac{d y}{d x} = \frac{1}{7} + \frac{4 x}{7}$