Calculus Examples

$f (t) = t^{- 5}$ ;, $(\frac{1}{2}, 32)$

Step 1

Find the first derivative and evaluate at $t = \frac{1}{2}$ and $y = 32$ to find the slope of the tangent line.

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Step 1.1

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

$- 5 t^{- 6}$

Step 1.2

Simplify.

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Step 1.2.1

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

$- 5 \frac{1}{t^{6}}$

Step 1.2.2

Combine $- 5$ and $\frac{1}{t^{6}}$ .

$\frac{- 5}{t^{6}}$

Step 1.2.3

Move the negative in front of the fraction.

$- \frac{5}{t^{6}}$

Step 1.3

Evaluate the derivative at $t = \frac{1}{2}$ .

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

Step 1.4

Simplify.

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Step 1.4.1

Simplify the denominator.

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Step 1.4.1.1

Apply the product rule to $\frac{1}{2}$ .

$- \frac{5}{\frac{1^{6}}{2^{6}}}$

Step 1.4.1.2

One to any power is one.

$- \frac{5}{\frac{1}{2^{6}}}$

Step 1.4.1.3

Raise $2$ to the power of $6$ .

$- \frac{5}{\frac{1}{64}}$

Step 1.4.2

Multiply the numerator by the reciprocal of the denominator.

$- (5 \cdot 64)$

Step 1.4.3

Multiply $- (5 \cdot 64)$ .

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Step 1.4.3.1

Multiply $5$ by $64$ .

$- 1 \cdot 320$

Step 1.4.3.2

Multiply $- 1$ by $320$ .

$- 320$

Step 2

Plug the slope and point values into the point-slope formula and solve for $y$ .

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Step 2.1

Use the slope $- 320$ and a given point $(\frac{1}{2}, 32)$ to substitute for $x_{1}$ and $y_{1}$ in the point-slope form $y - y_{1} = m (x - x_{1})$ , which is derived from the slope equation $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (32) = - 320 \cdot (x - (\frac{1}{2}))$

Step 2.2

Simplify the equation and keep it in point-slope form.

$y - 32 = - 320 \cdot (x - \frac{1}{2})$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $- 320 \cdot (x - \frac{1}{2})$ .

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Step 2.3.1.1

Rewrite.

$y - 32 = 0 + 0 - 320 \cdot (x - \frac{1}{2})$

Step 2.3.1.2

Simplify by adding zeros.

$y - 32 = - 320 \cdot (x - \frac{1}{2})$

Step 2.3.1.3

Apply the distributive property.

$y - 32 = - 320 x - 320 (- \frac{1}{2})$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Move the leading negative in $- \frac{1}{2}$ into the numerator.

$y - 32 = - 320 x - 320 (\frac{- 1}{2})$

Step 2.3.1.4.2

Factor $2$ out of $- 320$ .

$y - 32 = - 320 x + 2 (- 160) \frac{- 1}{2}$

Step 2.3.1.4.3

Cancel the common factor.

$y - 32 = - 320 x + 2 \cdot - 160 \frac{- 1}{2}$

Step 2.3.1.4.4

Rewrite the expression.

$y - 32 = - 320 x - 160 \cdot - 1$

Step 2.3.1.5

Multiply $- 160$ by $- 1$ .

$y - 32 = - 320 x + 160$

Step 2.3.2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.3.2.1

Add $32$ to both sides of the equation.

$y = - 320 x + 160 + 32$

Step 2.3.2.2

Add $160$ and $32$ .

$y = - 320 x + 192$

Step 3