Calculus Examples

$3 x^{2} + 24 x - y^{2} + 2 y = - 44$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (3 x^{2} + 24 x - y^{2} + 2 y) = \frac{d}{d x} (- 44)$

Step 2

Differentiate the left side of the equation.

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Step 2.1

By the Sum Rule, the derivative of $3 x^{2} + 24 x - y^{2} + 2 y$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [24 x] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [2 y]$ .

$\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [24 x] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [2 y]$

Step 2.2

Evaluate $\frac{d}{d x} [3 x^{2}]$ .

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Step 2.2.1

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

$3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [24 x] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [2 y]$

Step 2.2.2

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

$3 (2 x) + \frac{d}{d x} [24 x] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [2 y]$

Step 2.2.3

Multiply $2$ by $3$ .

$6 x + \frac{d}{d x} [24 x] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [2 y]$

Step 2.3

Evaluate $\frac{d}{d x} [24 x]$ .

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Step 2.3.1

Since $24$ is constant with respect to $x$ , the derivative of $24 x$ with respect to $x$ is $24 \frac{d}{d x} [x]$ .

$6 x + 24 \frac{d}{d x} [x] + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [2 y]$

Step 2.3.2

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

$6 x + 24 \cdot 1 + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [2 y]$

Step 2.3.3

Multiply $24$ by $1$ .

$6 x + 24 + \frac{d}{d x} [- y^{2}] + \frac{d}{d x} [2 y]$

Step 2.4

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

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Step 2.4.1

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

$6 x + 24 - \frac{d}{d x} [y^{2}] + \frac{d}{d x} [2 y]$

Step 2.4.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = y$ .

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Step 2.4.2.1

To apply the Chain Rule, set $u$ as $y$ .

$6 x + 24 - (\frac{d}{d u} [u^{2}] \frac{d}{d x} [y]) + \frac{d}{d x} [2 y]$

Step 2.4.2.2

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

$6 x + 24 - (2 u \frac{d}{d x} [y]) + \frac{d}{d x} [2 y]$

Step 2.4.2.3

Replace all occurrences of $u$ with $y$ .

$6 x + 24 - (2 y \frac{d}{d x} [y]) + \frac{d}{d x} [2 y]$

Step 2.4.3

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

$6 x + 24 - (2 y y') + \frac{d}{d x} [2 y]$

Step 2.4.4

Multiply $2$ by $- 1$ .

$6 x + 24 - 2 y y' + \frac{d}{d x} [2 y]$

Step 2.5

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

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Step 2.5.1

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

$6 x + 24 - 2 y y' + 2 \frac{d}{d x} [y]$

Step 2.5.2

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

$6 x + 24 - 2 y y' + 2 y'$

Step 2.6

Reorder terms.

$- 2 y y' + 6 x + 2 y' + 24$

Step 3

Since $- 44$ is constant with respect to $x$ , the derivative of $- 44$ with respect to $x$ is $0$ .

$0$

Step 4

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

$- 2 y y' + 6 x + 2 y' + 24 = 0$

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 $6 x$ from both sides of the equation.

$- 2 y y' + 2 y' + 24 = - 6 x$

Step 5.1.2

Subtract $24$ from both sides of the equation.

$- 2 y y' + 2 y' = - 6 x - 24$

Step 5.2

Factor $2 y'$ out of $- 2 y y' + 2 y'$ .

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Step 5.2.1

Factor $2 y'$ out of $- 2 y y'$ .

$2 y' (- y) + 2 y' = - 6 x - 24$

Step 5.2.2

Factor $2 y'$ out of $2 y'$ .

$2 y' (- y) + 2 y' \cdot 1 = - 6 x - 24$

Step 5.2.3

Factor $2 y'$ out of $2 y' (- y) + 2 y' \cdot 1$ .

$2 y' (- y + 1) = - 6 x - 24$

Step 5.3

Divide each term in $2 y' (- y + 1) = - 6 x - 24$ by $2 (- y + 1)$ and simplify.

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Step 5.3.1

Divide each term in $2 y' (- y + 1) = - 6 x - 24$ by $2 (- y + 1)$ .

$\frac{2 y' (- y + 1)}{2 (- y + 1)} = \frac{- 6 x}{2 (- y + 1)} + \frac{- 24}{2 (- y + 1)}$

Step 5.3.2

Simplify the left side.

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Step 5.3.2.1

Cancel the common factor of $2$ .

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Step 5.3.2.1.1

Cancel the common factor.

$\frac{2 y' (- y + 1)}{2 (- y + 1)} = \frac{- 6 x}{2 (- y + 1)} + \frac{- 24}{2 (- y + 1)}$

Step 5.3.2.1.2

Rewrite the expression.

$\frac{y' (- y + 1)}{- y + 1} = \frac{- 6 x}{2 (- y + 1)} + \frac{- 24}{2 (- y + 1)}$

Step 5.3.2.2

Cancel the common factor of $- y + 1$ .

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Step 5.3.2.2.1

Cancel the common factor.

$\frac{y' (- y + 1)}{- y + 1} = \frac{- 6 x}{2 (- y + 1)} + \frac{- 24}{2 (- y + 1)}$

Step 5.3.2.2.2

Divide $y'$ by $1$ .

$y' = \frac{- 6 x}{2 (- y + 1)} + \frac{- 24}{2 (- y + 1)}$

Step 5.3.3

Simplify the right side.

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Step 5.3.3.1

Simplify each term.

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Step 5.3.3.1.1

Cancel the common factor of $- 6$ and $2$ .

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Step 5.3.3.1.1.1

Factor $2$ out of $- 6 x$ .

$y' = \frac{2 (- 3 x)}{2 (- y + 1)} + \frac{- 24}{2 (- y + 1)}$

Step 5.3.3.1.1.2

Cancel the common factors.

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Step 5.3.3.1.1.2.1

Cancel the common factor.

$y' = \frac{2 (- 3 x)}{2 (- y + 1)} + \frac{- 24}{2 (- y + 1)}$

Step 5.3.3.1.1.2.2

Rewrite the expression.

$y' = \frac{- 3 x}{- y + 1} + \frac{- 24}{2 (- y + 1)}$

Step 5.3.3.1.2

Move the negative in front of the fraction.

$y' = - \frac{3 x}{- y + 1} + \frac{- 24}{2 (- y + 1)}$

Step 5.3.3.1.3

Cancel the common factor of $- 24$ and $2$ .

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Step 5.3.3.1.3.1

Factor $2$ out of $- 24$ .

$y' = - \frac{3 x}{- y + 1} + \frac{2 \cdot - 12}{2 (- y + 1)}$

Step 5.3.3.1.3.2

Cancel the common factors.

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Step 5.3.3.1.3.2.1

Cancel the common factor.

$y' = - \frac{3 x}{- y + 1} + \frac{2 \cdot - 12}{2 (- y + 1)}$

Step 5.3.3.1.3.2.2

Rewrite the expression.

$y' = - \frac{3 x}{- y + 1} + \frac{- 12}{- y + 1}$

Step 5.3.3.1.4

Move the negative in front of the fraction.

$y' = - \frac{3 x}{- y + 1} - \frac{12}{- y + 1}$

Step 5.3.3.2

Simplify terms.

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Step 5.3.3.2.1

Combine the numerators over the common denominator.

$y' = \frac{- 3 x - 12}{- y + 1}$

Step 5.3.3.2.2

Factor $3$ out of $- 3 x - 12$ .

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Step 5.3.3.2.2.1

Factor $3$ out of $- 3 x$ .

$y' = \frac{3 (- x) - 12}{- y + 1}$

Step 5.3.3.2.2.2

Factor $3$ out of $- 12$ .

$y' = \frac{3 (- x) + 3 (- 4)}{- y + 1}$

Step 5.3.3.2.2.3

Factor $3$ out of $3 (- x) + 3 (- 4)$ .

$y' = \frac{3 (- x - 4)}{- y + 1}$

Step 5.3.3.2.3

Factor $- 1$ out of $- x$ .

$y' = \frac{3 (- (x) - 4)}{- y + 1}$

Step 5.3.3.2.4

Rewrite $- 4$ as $- 1 (4)$ .

$y' = \frac{3 (- (x) - 1 (4))}{- y + 1}$

Step 5.3.3.2.5

Factor $- 1$ out of $- (x) - 1 (4)$ .

$y' = \frac{3 (- (x + 4))}{- y + 1}$

Step 5.3.3.2.6

Rewrite $- (x + 4)$ as $- 1 (x + 4)$ .

$y' = \frac{3 (- 1 (x + 4))}{- y + 1}$

Step 5.3.3.2.7

Factor $- 1$ out of $- y$ .

$y' = \frac{3 (- 1 (x + 4))}{- (y) + 1}$

Step 5.3.3.2.8

Rewrite $1$ as $- 1 (- 1)$ .

$y' = \frac{3 (- 1 (x + 4))}{- (y) - 1 (- 1)}$

Step 5.3.3.2.9

Factor $- 1$ out of $- (y) - 1 (- 1)$ .

$y' = \frac{3 (- 1 (x + 4))}{- (y - 1)}$

Step 5.3.3.2.10

Rewrite $- (y - 1)$ as $- 1 (y - 1)$ .

$y' = \frac{3 (- 1 (x + 4))}{- 1 (y - 1)}$

Step 5.3.3.2.11

Cancel the common factor.

$y' = \frac{3 (- 1 (x + 4))}{- 1 (y - 1)}$

Step 5.3.3.2.12

Rewrite the expression.

$y' = \frac{3 (x + 4)}{y - 1}$

Step 6

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

$\frac{d y}{d x} = \frac{3 (x + 4)}{y - 1}$