Calculus Examples

$f (x) = \frac{8 (x - t)}{x^{2}}$

Step 1

Since $8$ is constant with respect to $x$ , the derivative of $\frac{8 (x - t)}{x^{2}}$ with respect to $x$ is $8 \frac{d}{d x} [\frac{x - t}{x^{2}}]$ .

$8 \frac{d}{d x} [\frac{x - t}{x^{2}}]$

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

$8 \frac{x^{2} \frac{d}{d x} [x - t] - (x - t) \frac{d}{d x} [x^{2}]}{{(x^{2})}^{2}}$

Step 3

Differentiate.

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Step 3.1

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

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Step 3.1.1

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

$8 \frac{x^{2} \frac{d}{d x} [x - t] - (x - t) \frac{d}{d x} [x^{2}]}{x^{2 \cdot 2}}$

Step 3.1.2

Multiply $2$ by $2$ .

$8 \frac{x^{2} \frac{d}{d x} [x - t] - (x - t) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.2

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

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

Step 3.3

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

$8 \frac{x^{2} (1 + \frac{d}{d x} [- t]) - (x - t) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.4

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

$8 \frac{x^{2} (1 + 0) - (x - t) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.5

Simplify the expression.

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Step 3.5.1

Add $1$ and $0$ .

$8 \frac{x^{2} \cdot 1 - (x - t) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.5.2

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

$8 \frac{x^{2} - (x - t) \frac{d}{d x} [x^{2}]}{x^{4}}$

Step 3.6

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

$8 \frac{x^{2} - (x - t) (2 x)}{x^{4}}$

Step 3.7

Simplify with factoring out.

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Step 3.7.1

Multiply $2$ by $- 1$ .

$8 \frac{x^{2} - 2 (x - t) x}{x^{4}}$

Step 3.7.2

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

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Step 3.7.2.1

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

$8 \frac{x \cdot x - 2 (x - t) x}{x^{4}}$

Step 3.7.2.2

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

$8 \frac{x \cdot x + x (- 2 (x - t))}{x^{4}}$

Step 3.7.2.3

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

$8 \frac{x (x - 2 (x - t))}{x^{4}}$

Step 4

Cancel the common factors.

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Step 4.1

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

$8 \frac{x (x - 2 (x - t))}{x \cdot x^{3}}$

Step 4.2

Cancel the common factor.

$8 \frac{x (x - 2 (x - t))}{x \cdot x^{3}}$

Step 4.3

Rewrite the expression.

$8 \frac{x - 2 (x - t)}{x^{3}}$

Step 5

Combine $8$ and $\frac{x - 2 (x - t)}{x^{3}}$ .

$\frac{8 (x - 2 (x - t))}{x^{3}}$

Step 6

Simplify.

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Step 6.1

Apply the distributive property.

$\frac{8 (x - 2 x - 2 (- t))}{x^{3}}$

Step 6.2

Apply the distributive property.

$\frac{8 x + 8 (- 2 x) + 8 (- 2 (- t))}{x^{3}}$

Step 6.3

Simplify the numerator.

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Step 6.3.1

Simplify each term.

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Step 6.3.1.1

Multiply $- 2$ by $8$ .

$\frac{8 x - 16 x + 8 (- 2 (- t))}{x^{3}}$

Step 6.3.1.2

Multiply $- 1$ by $- 2$ .

$\frac{8 x - 16 x + 8 (2 t)}{x^{3}}$

Step 6.3.1.3

Multiply $2$ by $8$ .

$\frac{8 x - 16 x + 16 t}{x^{3}}$

Step 6.3.2

Subtract $16 x$ from $8 x$ .

$\frac{- 8 x + 16 t}{x^{3}}$

Step 6.4

Reorder terms.

$\frac{16 t - 8 x}{x^{3}}$

Step 6.5

Factor $8$ out of $16 t - 8 x$ .

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Step 6.5.1

Factor $8$ out of $16 t$ .

$\frac{8 (2 t) - 8 x}{x^{3}}$

Step 6.5.2

Factor $8$ out of $- 8 x$ .

$\frac{8 (2 t) + 8 (- x)}{x^{3}}$

Step 6.5.3

Factor $8$ out of $8 (2 t) + 8 (- x)$ .

$\frac{8 (2 t - x)}{x^{3}}$