Calculus Examples

$4 x + 9 y = 7$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (4 x + 9 y) = \frac{d}{d x} (7)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $4 x + 9 y$ with respect to $x$ is $\frac{d}{d x} [4 x] + \frac{d}{d x} [9 y]$ .

$\frac{d}{d x} [4 x] + \frac{d}{d x} [9 y]$

Step 2.2

Evaluate $\frac{d}{d x} [4 x]$ .

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Step 2.2.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x] + \frac{d}{d x} [9 y]$

Step 2.2.2

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

$4 \cdot 1 + \frac{d}{d x} [9 y]$

Step 2.2.3

Multiply $4$ by $1$ .

$4 + \frac{d}{d x} [9 y]$

Step 2.3

Evaluate $\frac{d}{d x} [9 y]$ .

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Step 2.3.1

Since $9$ is constant with respect to $x$ , the derivative of $9 y$ with respect to $x$ is $9 \frac{d}{d x} [y]$ .

$4 + 9 \frac{d}{d x} [y]$

Step 2.3.2

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

$4 + 9 y'$

Step 2.4

Reorder terms.

$9 y' + 4$

Step 3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$0$

Step 4

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

$9 y' + 4 = 0$

Step 5

Solve for $y'$ .

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Step 5.1

Subtract $4$ from both sides of the equation.

$9 y' = - 4$

Step 5.2

Divide each term in $9 y' = - 4$ by $9$ and simplify.

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Step 5.2.1

Divide each term in $9 y' = - 4$ by $9$ .

$\frac{9 y'}{9} = \frac{- 4}{9}$

Step 5.2.2

Simplify the left side.

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Step 5.2.2.1

Cancel the common factor of $9$ .

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Step 5.2.2.1.1

Cancel the common factor.

$\frac{9 y'}{9} = \frac{- 4}{9}$

Step 5.2.2.1.2

Divide $y'$ by $1$ .

$y' = \frac{- 4}{9}$

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.

$y' = - \frac{4}{9}$

Step 6

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

$\frac{d y}{d x} = - \frac{4}{9}$

Step 7

Set $\frac{d y}{d x} = 0$ then solve for $x$ in terms of $y$ .

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Step 7.1

Set the numerator equal to zero.

$4 = 0$

Step 7.2

Since $4 \neq 0$ , there are no solutions.

No solution

Step 8

No points that set $\frac{d y}{d x} = 0$ are on the real number plane.

No Points

Step 9