Calculus Examples

$p (x) = \frac{1040}{0.4 x^{2} + 8}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Since $1040$ is constant with respect to $x$ , the derivative of $\frac{1040}{0.4 x^{2} + 8}$ with respect to $x$ is $1040 \frac{d}{d x} [\frac{1}{0.4 x^{2} + 8}]$ .

$1040 \frac{d}{d x} [\frac{1}{0.4 x^{2} + 8}]$

Step 1.2

Rewrite $\frac{1}{0.4 x^{2} + 8}$ as ${(0.4 x^{2} + 8)}^{- 1}$ .

$1040 \frac{d}{d x} [{(0.4 x^{2} + 8)}^{- 1}]$

Step 2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{- 1}$ and $g (x) = 0.4 x^{2} + 8$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $0.4 x^{2} + 8$ .

$1040 (\frac{d}{d u} [u^{- 1}] \frac{d}{d x} [0.4 x^{2} + 8])$

Step 2.2

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

$1040 (- u^{- 2} \frac{d}{d x} [0.4 x^{2} + 8])$

Step 2.3

Replace all occurrences of $u$ with $0.4 x^{2} + 8$ .

$1040 (- {(0.4 x^{2} + 8)}^{- 2} \frac{d}{d x} [0.4 x^{2} + 8])$

Step 3

Differentiate.

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Step 3.1

Multiply $- 1$ by $1040$ .

$- 1040 ({(0.4 x^{2} + 8)}^{- 2} \frac{d}{d x} [0.4 x^{2} + 8])$

Step 3.2

By the Sum Rule, the derivative of $0.4 x^{2} + 8$ with respect to $x$ is $\frac{d}{d x} [0.4 x^{2}] + \frac{d}{d x} [8]$ .

$- 1040 {(0.4 x^{2} + 8)}^{- 2} (\frac{d}{d x} [0.4 x^{2}] + \frac{d}{d x} [8])$

Step 3.3

Since $0.4$ is constant with respect to $x$ , the derivative of $0.4 x^{2}$ with respect to $x$ is $0.4 \frac{d}{d x} [x^{2}]$ .

$- 1040 {(0.4 x^{2} + 8)}^{- 2} (0.4 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [8])$

Step 3.4

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

$- 1040 {(0.4 x^{2} + 8)}^{- 2} (0.4 (2 x) + \frac{d}{d x} [8])$

Step 3.5

Multiply $2$ by $0.4$ .

$- 1040 {(0.4 x^{2} + 8)}^{- 2} (0.8 x + \frac{d}{d x} [8])$

Step 3.6

Since $8$ is constant with respect to $x$ , the derivative of $8$ with respect to $x$ is $0$ .

$- 1040 {(0.4 x^{2} + 8)}^{- 2} (0.8 x + 0)$

Step 3.7

Simplify the expression.

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Step 3.7.1

Add $0.8 x$ and $0$ .

$- 1040 {(0.4 x^{2} + 8)}^{- 2} (0.8 x)$

Step 3.7.2

Multiply $0.8$ by $- 1040$ .

$- 832 {(0.4 x^{2} + 8)}^{- 2} x$

Step 4

Simplify.

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Step 4.1

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

$- 832 \frac{1}{{(0.4 x^{2} + 8)}^{2}} x$

Step 4.2

Combine terms.

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Step 4.2.1

Combine $- 832$ and $\frac{1}{{(0.4 x^{2} + 8)}^{2}}$ .

$\frac{- 832}{{(0.4 x^{2} + 8)}^{2}} x$

Step 4.2.2

Move the negative in front of the fraction.

$- \frac{832}{{(0.4 x^{2} + 8)}^{2}} x$

Step 4.2.3

Combine $x$ and $\frac{832}{{(0.4 x^{2} + 8)}^{2}}$ .

$- \frac{x \cdot 832}{{(0.4 x^{2} + 8)}^{2}}$

Step 4.2.4

Move $832$ to the left of $x$ .

$- \frac{832 x}{{(0.4 x^{2} + 8)}^{2}}$

Step 4.3

Simplify the denominator.

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Step 4.3.1

Factor $0.4$ out of $0.4 x^{2} + 8$ .

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Step 4.3.1.1

Factor $0.4$ out of $0.4 x^{2}$ .

$- \frac{832 x}{{(0.4 (x^{2}) + 8)}^{2}}$

Step 4.3.1.2

Factor $0.4$ out of $8$ .

$- \frac{832 x}{{(0.4 x^{2} + 0.4 \cdot 20)}^{2}}$

Step 4.3.1.3

Factor $0.4$ out of $0.4 x^{2} + 0.4 \cdot 20$ .

$- \frac{832 x}{{(0.4 (x^{2} + 20))}^{2}}$

Step 4.3.2

Apply the product rule to $0.4 (x^{2} + 20)$ .

$- \frac{832 x}{{0.4}^{2} {(x^{2} + 20)}^{2}}$

Step 4.3.3

Raise $0.4$ to the power of $2$ .

$- \frac{832 x}{0.16 {(x^{2} + 20)}^{2}}$

Step 4.4

Factor $832$ out of $832 x$ .

$- \frac{832 (x)}{0.16 {(x^{2} + 20)}^{2}}$

Step 4.5

Factor $0.16$ out of $0.16 {(x^{2} + 20)}^{2}$ .

$- \frac{832 (x)}{0.16 ({(x^{2} + 20)}^{2})}$

Step 4.6

Separate fractions.

$- (\frac{832}{0.16} \cdot \frac{x}{{(x^{2} + 20)}^{2}})$

Step 4.7

Divide $832$ by $0.16$ .

$- (5200 \frac{x}{{(x^{2} + 20)}^{2}})$

Step 4.8

Combine $5200$ and $\frac{x}{{(x^{2} + 20)}^{2}}$ .

$- \frac{5200 x}{{(x^{2} + 20)}^{2}}$