Calculus Examples

$y = {(4 x^{2} - 14)}^{- 10}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 10}$ and $g (x) = 4 x^{2} - 14$ .

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Step 1.1

To apply the Chain Rule, set $u$ as $4 x^{2} - 14$ .

$\frac{d}{d u} [u^{- 10}] \frac{d}{d x} [4 x^{2} - 14]$

Step 1.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = - 10$ .

$- 10 u^{- 11} \frac{d}{d x} [4 x^{2} - 14]$

Step 1.3

Replace all occurrences of $u$ with $4 x^{2} - 14$ .

$- 10 {(4 x^{2} - 14)}^{- 11} \frac{d}{d x} [4 x^{2} - 14]$

Step 2

By the Sum Rule, the derivative of $4 x^{2} - 14$ with respect to $x$ is $\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 14]$ .

$- 10 {(4 x^{2} - 14)}^{- 11} (\frac{d}{d x} [4 x^{2}] + \frac{d}{d x} [- 14])$

Step 3

Evaluate $\frac{d}{d x} [4 x^{2}]$ .

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Step 3.1

Since $4$ is constant with respect to $x$ , the derivative of $4 x^{2}$ with respect to $x$ is $4 \frac{d}{d x} [x^{2}]$ .

$- 10 {(4 x^{2} - 14)}^{- 11} (4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 14])$

Step 3.2

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

$- 10 {(4 x^{2} - 14)}^{- 11} (4 (2 x) + \frac{d}{d x} [- 14])$

Step 3.3

Multiply $2$ by $4$ .

$- 10 {(4 x^{2} - 14)}^{- 11} (8 x + \frac{d}{d x} [- 14])$

Step 4

Differentiate using the Constant Rule.

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Step 4.1

Since $- 14$ is constant with respect to $x$ , the derivative of $- 14$ with respect to $x$ is $0$ .

$- 10 {(4 x^{2} - 14)}^{- 11} (8 x + 0)$

Step 4.2

Add $8 x$ and $0$ .

$- 10 {(4 x^{2} - 14)}^{- 11} (8 x)$

Step 5

Simplify.

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Step 5.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$- 10 \frac{1}{{(4 x^{2} - 14)}^{11}} (8 x)$

Step 5.2

Combine terms.

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Step 5.2.1

Combine $- 10$ and $\frac{1}{{(4 x^{2} - 14)}^{11}}$ .

$\frac{- 10}{{(4 x^{2} - 14)}^{11}} (8 x)$

Step 5.2.2

Move the negative in front of the fraction.

$- \frac{10}{{(4 x^{2} - 14)}^{11}} (8 x)$

Step 5.2.3

Multiply $8$ by $- 1$ .

$- 8 \frac{10}{{(4 x^{2} - 14)}^{11}} x$

Step 5.2.4

Combine $- 8$ and $\frac{10}{{(4 x^{2} - 14)}^{11}}$ .

$\frac{- 8 \cdot 10}{{(4 x^{2} - 14)}^{11}} x$

Step 5.2.5

Multiply $- 8$ by $10$ .

$\frac{- 80}{{(4 x^{2} - 14)}^{11}} x$

Step 5.2.6

Combine $\frac{- 80}{{(4 x^{2} - 14)}^{11}}$ and $x$ .

$\frac{- 80 x}{{(4 x^{2} - 14)}^{11}}$

Step 5.2.7

Move the negative in front of the fraction.

$- \frac{80 x}{{(4 x^{2} - 14)}^{11}}$

Step 5.3

Simplify the denominator.

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Step 5.3.1

Factor $2$ out of $4 x^{2} - 14$ .

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Step 5.3.1.1

Factor $2$ out of $4 x^{2}$ .

$- \frac{80 x}{{(2 (2 x^{2}) - 14)}^{11}}$

Step 5.3.1.2

Factor $2$ out of $- 14$ .

$- \frac{80 x}{{(2 (2 x^{2}) + 2 (- 7))}^{11}}$

Step 5.3.1.3

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

$- \frac{80 x}{{(2 (2 x^{2} - 7))}^{11}}$

Step 5.3.2

Apply the product rule to $2 (2 x^{2} - 7)$ .

$- \frac{80 x}{2^{11} {(2 x^{2} - 7)}^{11}}$

Step 5.3.3

Raise $2$ to the power of $11$ .

$- \frac{80 x}{2048 {(2 x^{2} - 7)}^{11}}$

Step 5.4

Cancel the common factor of $80$ and $2048$ .

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Step 5.4.1

Factor $16$ out of $80 x$ .

$- \frac{16 (5 x)}{2048 {(2 x^{2} - 7)}^{11}}$

Step 5.4.2

Cancel the common factors.

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Step 5.4.2.1

Factor $16$ out of $2048 {(2 x^{2} - 7)}^{11}$ .

$- \frac{16 (5 x)}{16 (128 {(2 x^{2} - 7)}^{11})}$

Step 5.4.2.2

Cancel the common factor.

$- \frac{16 (5 x)}{16 (128 {(2 x^{2} - 7)}^{11})}$

Step 5.4.2.3

Rewrite the expression.

$- \frac{5 x}{128 {(2 x^{2} - 7)}^{11}}$