Calculus Examples

$y = \frac{x^{5}}{{(x^{2} - 6)}^{6}}$

Step 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 $f (x) = x^{5}$ and $g (x) = {(x^{2} - 6)}^{6}$ .

$\frac{{(x^{2} - 6)}^{6} \frac{d}{d x} [x^{5}] - x^{5} \frac{d}{d x} [{(x^{2} - 6)}^{6}]}{{({(x^{2} - 6)}^{6})}^{2}}$

Step 2

Differentiate using the Power Rule.

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Step 2.1

Multiply the exponents in ${({(x^{2} - 6)}^{6})}^{2}$ .

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Step 2.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{{(x^{2} - 6)}^{6} \frac{d}{d x} [x^{5}] - x^{5} \frac{d}{d x} [{(x^{2} - 6)}^{6}]}{{(x^{2} - 6)}^{6 \cdot 2}}$

Step 2.1.2

Multiply $6$ by $2$ .

$\frac{{(x^{2} - 6)}^{6} \frac{d}{d x} [x^{5}] - x^{5} \frac{d}{d x} [{(x^{2} - 6)}^{6}]}{{(x^{2} - 6)}^{12}}$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$\frac{{(x^{2} - 6)}^{6} (5 x^{4}) - x^{5} \frac{d}{d x} [{(x^{2} - 6)}^{6}]}{{(x^{2} - 6)}^{12}}$

Step 2.3

Move $5$ to the left of ${(x^{2} - 6)}^{6}$ .

$\frac{5 \cdot {(x^{2} - 6)}^{6} x^{4} - x^{5} \frac{d}{d x} [{(x^{2} - 6)}^{6}]}{{(x^{2} - 6)}^{12}}$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{6}$ and $g (x) = x^{2} - 6$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x^{2} - 6$ .

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - x^{5} (\frac{d}{d u} [u^{6}] \frac{d}{d x} [x^{2} - 6])}{{(x^{2} - 6)}^{12}}$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = 6$ .

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - x^{5} (6 u^{5} \frac{d}{d x} [x^{2} - 6])}{{(x^{2} - 6)}^{12}}$

Step 3.3

Replace all occurrences of $u$ with $x^{2} - 6$ .

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - x^{5} (6 {(x^{2} - 6)}^{5} \frac{d}{d x} [x^{2} - 6])}{{(x^{2} - 6)}^{12}}$

Step 4

Differentiate.

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Step 4.1

Multiply $6$ by $- 1$ .

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - 6 x^{5} ({(x^{2} - 6)}^{5} \frac{d}{d x} [x^{2} - 6])}{{(x^{2} - 6)}^{12}}$

Step 4.2

By the Sum Rule, the derivative of $x^{2} - 6$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6]$ .

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - 6 x^{5} ({(x^{2} - 6)}^{5} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 6]))}{{(x^{2} - 6)}^{12}}$

Step 4.3

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

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - 6 x^{5} ({(x^{2} - 6)}^{5} (2 x + \frac{d}{d x} [- 6]))}{{(x^{2} - 6)}^{12}}$

Step 4.4

Since $- 6$ is constant with respect to $x$ , the derivative of $- 6$ with respect to $x$ is $0$ .

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - 6 x^{5} ({(x^{2} - 6)}^{5} (2 x + 0))}{{(x^{2} - 6)}^{12}}$

Step 4.5

Simplify the expression.

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Step 4.5.1

Add $2 x$ and $0$ .

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - 6 x^{5} ({(x^{2} - 6)}^{5} (2 x))}{{(x^{2} - 6)}^{12}}$

Step 4.5.2

Move $2$ to the left of ${(x^{2} - 6)}^{5}$ .

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - 6 x^{5} (2 \cdot {(x^{2} - 6)}^{5} x)}{{(x^{2} - 6)}^{12}}$

Step 4.5.3

Multiply $2$ by $- 6$ .

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - 12 x^{5} ({(x^{2} - 6)}^{5} x)}{{(x^{2} - 6)}^{12}}$

Step 5

Raise $x$ to the power of $1$ .

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - 12 (x^{1} x^{5}) {(x^{2} - 6)}^{5}}{{(x^{2} - 6)}^{12}}$

Step 6

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - 12 x^{1 + 5} {(x^{2} - 6)}^{5}}{{(x^{2} - 6)}^{12}}$

Step 7

Add $1$ and $5$ .

$\frac{5 {(x^{2} - 6)}^{6} x^{4} - 12 x^{6} {(x^{2} - 6)}^{5}}{{(x^{2} - 6)}^{12}}$

Step 8

Simplify.

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Step 8.1

Simplify the numerator.

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Step 8.1.1

Factor ${(x^{2} - 6)}^{5} x^{4}$ out of $5 {(x^{2} - 6)}^{6} x^{4} - 12 x^{6} {(x^{2} - 6)}^{5}$ .

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Step 8.1.1.1

Factor ${(x^{2} - 6)}^{5} x^{4}$ out of $5 {(x^{2} - 6)}^{6} x^{4}$ .

$\frac{{(x^{2} - 6)}^{5} x^{4} (5 (x^{2} - 6)) - 12 x^{6} {(x^{2} - 6)}^{5}}{{(x^{2} - 6)}^{12}}$

Step 8.1.1.2

Factor ${(x^{2} - 6)}^{5} x^{4}$ out of $- 12 x^{6} {(x^{2} - 6)}^{5}$ .

$\frac{{(x^{2} - 6)}^{5} x^{4} (5 (x^{2} - 6)) + {(x^{2} - 6)}^{5} x^{4} (- 12 x^{2})}{{(x^{2} - 6)}^{12}}$

Step 8.1.1.3

Factor ${(x^{2} - 6)}^{5} x^{4}$ out of ${(x^{2} - 6)}^{5} x^{4} (5 (x^{2} - 6)) + {(x^{2} - 6)}^{5} x^{4} (- 12 x^{2})$ .

$\frac{{(x^{2} - 6)}^{5} x^{4} (5 (x^{2} - 6) - 12 x^{2})}{{(x^{2} - 6)}^{12}}$

Step 8.1.2

Apply the distributive property.

$\frac{{(x^{2} - 6)}^{5} x^{4} (5 x^{2} + 5 \cdot - 6 - 12 x^{2})}{{(x^{2} - 6)}^{12}}$

Step 8.1.3

Multiply $5$ by $- 6$ .

$\frac{{(x^{2} - 6)}^{5} x^{4} (5 x^{2} - 30 - 12 x^{2})}{{(x^{2} - 6)}^{12}}$

Step 8.1.4

Subtract $12 x^{2}$ from $5 x^{2}$ .

$\frac{{(x^{2} - 6)}^{5} x^{4} (- 7 x^{2} - 30)}{{(x^{2} - 6)}^{12}}$

Step 8.2

Cancel the common factor of ${(x^{2} - 6)}^{5}$ and ${(x^{2} - 6)}^{12}$ .

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Step 8.2.1

Factor ${(x^{2} - 6)}^{5}$ out of ${(x^{2} - 6)}^{5} x^{4} (- 7 x^{2} - 30)$ .

$\frac{{(x^{2} - 6)}^{5} (x^{4} (- 7 x^{2} - 30))}{{(x^{2} - 6)}^{12}}$

Step 8.2.2

Cancel the common factors.

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Step 8.2.2.1

Factor ${(x^{2} - 6)}^{5}$ out of ${(x^{2} - 6)}^{12}$ .

$\frac{{(x^{2} - 6)}^{5} (x^{4} (- 7 x^{2} - 30))}{{(x^{2} - 6)}^{5} {(x^{2} - 6)}^{7}}$

Step 8.2.2.2

Cancel the common factor.

$\frac{{(x^{2} - 6)}^{5} (x^{4} (- 7 x^{2} - 30))}{{(x^{2} - 6)}^{5} {(x^{2} - 6)}^{7}}$

Step 8.2.2.3

Rewrite the expression.

$\frac{x^{4} (- 7 x^{2} - 30)}{{(x^{2} - 6)}^{7}}$

Step 8.3

Factor $- 1$ out of $- 7 x^{2}$ .

$\frac{x^{4} (- (7 x^{2}) - 30)}{{(x^{2} - 6)}^{7}}$

Step 8.4

Rewrite $- 30$ as $- 1 (30)$ .

$\frac{x^{4} (- (7 x^{2}) - 1 (30))}{{(x^{2} - 6)}^{7}}$

Step 8.5

Factor $- 1$ out of $- (7 x^{2}) - 1 (30)$ .

$\frac{x^{4} (- (7 x^{2} + 30))}{{(x^{2} - 6)}^{7}}$

Step 8.6

Rewrite $- (7 x^{2} + 30)$ as $- 1 (7 x^{2} + 30)$ .

$\frac{x^{4} (- 1 (7 x^{2} + 30))}{{(x^{2} - 6)}^{7}}$

Step 8.7

Move the negative in front of the fraction.

$- \frac{x^{4} (7 x^{2} + 30)}{{(x^{2} - 6)}^{7}}$