Calculus Examples

$f (x) = \frac{x^{2}}{3 (1 - x)}$ ; $x = 0.4$

Step 1

Find the derivative.

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Step 1.1

Since $\frac{1}{3}$ is constant with respect to $x$ , the derivative of $\frac{x^{2}}{3 (1 - x)}$ with respect to $x$ is $\frac{1}{3} \frac{d}{d x} [\frac{x^{2}}{1 - x}]$ .

$\frac{1}{3} \frac{d}{d x} [\frac{x^{2}}{1 - x}]$

Step 1.2

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 $f (x) = x^{2}$ and $g (x) = 1 - x$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

Move $2$ to the left of $1 - x$ .

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.3.6

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

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

Step 1.3.7

Multiply.

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Step 1.3.7.1

Multiply $- 1$ by $- 1$ .

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

Step 1.3.7.2

Multiply $x^{2}$ by $1$ .

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

Step 1.3.8

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

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

Step 1.3.9

Combine fractions.

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Step 1.3.9.1

Multiply $x^{2}$ by $1$ .

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

Step 1.3.9.2

Multiply $\frac{1}{3}$ by $\frac{2 (1 - x) x + x^{2}}{{(1 - x)}^{2}}$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Apply the distributive property.

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

Step 1.4.2

Apply the distributive property.

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

Step 1.4.3

Simplify the numerator.

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Step 1.4.3.1

Simplify each term.

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Step 1.4.3.1.1

Multiply $2$ by $1$ .

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

Step 1.4.3.1.2

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

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Step 1.4.3.1.2.1

Move $x$ .

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

Step 1.4.3.1.2.2

Multiply $x$ by $x$ .

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

Step 1.4.3.1.3

Multiply $- 1$ by $2$ .

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

Step 1.4.3.2

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

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

Step 1.4.4

Reorder terms.

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

Step 1.4.5

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

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Step 1.4.5.1

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

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

Step 1.4.5.2

Factor $x$ out of $2 x$ .

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

Step 1.4.5.3

Factor $x$ out of $x (- x) + x \cdot 2$ .

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

Step 1.4.6

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

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

Step 1.4.7

Rewrite $2$ as $- 1 (- 2)$ .

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

Step 1.4.8

Factor $- 1$ out of $- (x) - 1 (- 2)$ .

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

Step 1.4.9

Rewrite $- (x - 2)$ as $- 1 (x - 2)$ .

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

Step 1.4.10

Move the negative in front of the fraction.

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

Step 2

Replace the variable $x$ with $0.4$ in the expression.

$f' (0.4) = - \frac{(0.4) ((0.4) - 2)}{3 {(1 - (0.4))}^{2}}$

Step 3

Multiply $0.4$ by $0.4 - 2$ .

$f' (0.4) = - \frac{- 0.64}{3 {(1 - (0.4))}^{2}}$

Step 4

Simplify the denominator.

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Step 4.1

Multiply $- 1$ by $0.4$ .

$f' (0.4) = - \frac{- 0.64}{3 {(1 - 0.4)}^{2}}$

Step 4.2

Subtract $0.4$ from $1$ .

$f' (0.4) = - \frac{- 0.64}{3 \cdot {0.6}^{2}}$

Step 4.3

Raise $0.6$ to the power of $2$ .

$f' (0.4) = - \frac{- 0.64}{3 \cdot 0.36}$

Step 5

Simplify the expression.

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Step 5.1

Multiply $3$ by $0.36$ .

$f' (0.4) = - \frac{- 0.64}{1.08}$

Step 5.2

Divide $- 0.64$ by $1.08$ .

$f' (0.4) = 0.\overline{592}$

Step 5.3

Multiply $- 1$ by $- 0.\overline{592}$ .

$f' (0.4) = 0.\overline{592}$