Calculus Examples

$y = \frac{1 - 3 x}{x^{2} + 7}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (\frac{1 - 3 x}{x^{2} + 7})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Differentiate using the Quotient Rule which states that $\frac{d}{d x} [\frac{f (x)}{g (x)}]$ is $\frac{g (x) \frac{d}{d x} [f (x)] - f (x) \frac{d}{d x} [g (x)]}{{g (x)}^{2}}$ where $f (x) = 1 - 3 x$ and $g (x) = x^{2} + 7$ .

$\frac{(x^{2} + 7) \frac{d}{d x} [1 - 3 x] - (1 - 3 x) \frac{d}{d x} [x^{2} + 7]}{{(x^{2} + 7)}^{2}}$

Step 3.2

Differentiate.

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Step 3.2.1

By the Sum Rule, the derivative of $1 - 3 x$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- 3 x]$ .

$\frac{(x^{2} + 7) (\frac{d}{d x} [1] + \frac{d}{d x} [- 3 x]) - (1 - 3 x) \frac{d}{d x} [x^{2} + 7]}{{(x^{2} + 7)}^{2}}$

Step 3.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$\frac{(x^{2} + 7) (0 + \frac{d}{d x} [- 3 x]) - (1 - 3 x) \frac{d}{d x} [x^{2} + 7]}{{(x^{2} + 7)}^{2}}$

Step 3.2.3

Add $0$ and $\frac{d}{d x} [- 3 x]$ .

$\frac{(x^{2} + 7) \frac{d}{d x} [- 3 x] - (1 - 3 x) \frac{d}{d x} [x^{2} + 7]}{{(x^{2} + 7)}^{2}}$

Step 3.2.4

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

$\frac{(x^{2} + 7) (- 3 \frac{d}{d x} [x]) - (1 - 3 x) \frac{d}{d x} [x^{2} + 7]}{{(x^{2} + 7)}^{2}}$

Step 3.2.5

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

$\frac{(x^{2} + 7) (- 3 \cdot 1) - (1 - 3 x) \frac{d}{d x} [x^{2} + 7]}{{(x^{2} + 7)}^{2}}$

Step 3.2.6

Simplify the expression.

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Step 3.2.6.1

Multiply $- 3$ by $1$ .

$\frac{(x^{2} + 7) \cdot - 3 - (1 - 3 x) \frac{d}{d x} [x^{2} + 7]}{{(x^{2} + 7)}^{2}}$

Step 3.2.6.2

Move $- 3$ to the left of $x^{2} + 7$ .

$\frac{- 3 \cdot (x^{2} + 7) - (1 - 3 x) \frac{d}{d x} [x^{2} + 7]}{{(x^{2} + 7)}^{2}}$

Step 3.2.7

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

$\frac{- 3 (x^{2} + 7) - (1 - 3 x) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [7])}{{(x^{2} + 7)}^{2}}$

Step 3.2.8

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

$\frac{- 3 (x^{2} + 7) - (1 - 3 x) (2 x + \frac{d}{d x} [7])}{{(x^{2} + 7)}^{2}}$

Step 3.2.9

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

$\frac{- 3 (x^{2} + 7) - (1 - 3 x) (2 x + 0)}{{(x^{2} + 7)}^{2}}$

Step 3.2.10

Simplify the expression.

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Step 3.2.10.1

Add $2 x$ and $0$ .

$\frac{- 3 (x^{2} + 7) - (1 - 3 x) (2 x)}{{(x^{2} + 7)}^{2}}$

Step 3.2.10.2

Multiply $2$ by $- 1$ .

$\frac{- 3 (x^{2} + 7) - 2 (1 - 3 x) x}{{(x^{2} + 7)}^{2}}$

Step 3.3

Simplify.

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Step 3.3.1

Apply the distributive property.

$\frac{- 3 x^{2} - 3 \cdot 7 - 2 (1 - 3 x) x}{{(x^{2} + 7)}^{2}}$

Step 3.3.2

Apply the distributive property.

$\frac{- 3 x^{2} - 3 \cdot 7 + (- 2 \cdot 1 - 2 (- 3 x)) x}{{(x^{2} + 7)}^{2}}$

Step 3.3.3

Apply the distributive property.

$\frac{- 3 x^{2} - 3 \cdot 7 - 2 \cdot 1 x - 2 (- 3 x) x}{{(x^{2} + 7)}^{2}}$

Step 3.3.4

Simplify the numerator.

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Step 3.3.4.1

Simplify each term.

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Step 3.3.4.1.1

Multiply $- 3$ by $7$ .

$\frac{- 3 x^{2} - 21 - 2 \cdot 1 x - 2 (- 3 x) x}{{(x^{2} + 7)}^{2}}$

Step 3.3.4.1.2

Multiply $- 2$ by $1$ .

$\frac{- 3 x^{2} - 21 - 2 x - 2 (- 3 x) x}{{(x^{2} + 7)}^{2}}$

Step 3.3.4.1.3

Multiply $x$ by $x$ by adding the exponents.

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Step 3.3.4.1.3.1

Move $x$ .

$\frac{- 3 x^{2} - 21 - 2 x - 2 (- 3 (x \cdot x))}{{(x^{2} + 7)}^{2}}$

Step 3.3.4.1.3.2

Multiply $x$ by $x$ .

$\frac{- 3 x^{2} - 21 - 2 x - 2 (- 3 x^{2})}{{(x^{2} + 7)}^{2}}$

Step 3.3.4.1.4

Multiply $- 3$ by $- 2$ .

$\frac{- 3 x^{2} - 21 - 2 x + 6 x^{2}}{{(x^{2} + 7)}^{2}}$

Step 3.3.4.2

Add $- 3 x^{2}$ and $6 x^{2}$ .

$\frac{3 x^{2} - 21 - 2 x}{{(x^{2} + 7)}^{2}}$

Step 3.3.5

Reorder terms.

$\frac{3 x^{2} - 2 x - 21}{{(x^{2} + 7)}^{2}}$

Step 3.3.6

Factor by grouping.

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Step 3.3.6.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 21 = - 63$ and whose sum is $b = - 2$ .

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Step 3.3.6.1.1

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

$\frac{3 x^{2} - 2 (x) - 21}{{(x^{2} + 7)}^{2}}$

Step 3.3.6.1.2

Rewrite $- 2$ as $7$ plus $- 9$

$\frac{3 x^{2} + (7 - 9) x - 21}{{(x^{2} + 7)}^{2}}$

Step 3.3.6.1.3

Apply the distributive property.

$\frac{3 x^{2} + 7 x - 9 x - 21}{{(x^{2} + 7)}^{2}}$

Step 3.3.6.2

Factor out the greatest common factor from each group.

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Step 3.3.6.2.1

Group the first two terms and the last two terms.

$\frac{(3 x^{2} + 7 x) - 9 x - 21}{{(x^{2} + 7)}^{2}}$

Step 3.3.6.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{x (3 x + 7) - 3 (3 x + 7)}{{(x^{2} + 7)}^{2}}$

Step 3.3.6.3

Factor the polynomial by factoring out the greatest common factor, $3 x + 7$ .

$\frac{(3 x + 7) (x - 3)}{{(x^{2} + 7)}^{2}}$

Step 4

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

$y' = \frac{(3 x + 7) (x - 3)}{{(x^{2} + 7)}^{2}}$

Step 5

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

$\frac{d y}{d x} = \frac{(3 x + 7) (x - 3)}{{(x^{2} + 7)}^{2}}$