Calculus Examples

$f (x) = \tan (x)$ , $(- \frac{π}{4}, - 1)$

Step 1

Find the first derivative and evaluate at $x = - \frac{π}{4}$ and $y = - 1$ to find the slope of the tangent line.

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Step 1.1

The derivative of $\tan (x)$ with respect to $x$ is $\sec^{2} (x)$ .

$\sec^{2} (x)$

Step 1.2

Evaluate the derivative at $x = - \frac{π}{4}$ .

$\sec^{2} (- \frac{π}{4})$

Step 1.3

Simplify.

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Step 1.3.1

Add full rotations of $2 π$ until the angle is greater than or equal to $0$ and less than $2 π$ .

$\sec^{2} (\frac{7 π}{4})$

Step 1.3.2

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant.

$\sec^{2} (\frac{π}{4})$

Step 1.3.3

The exact value of $\sec (\frac{π}{4})$ is $\frac{2}{\sqrt{2}}$ .

${(\frac{2}{\sqrt{2}})}^{2}$

Step 1.3.4

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2}$

Step 1.3.5

Combine and simplify the denominator.

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Step 1.3.5.1

Multiply $\frac{2}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

${(\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2}$

Step 1.3.5.2

Raise $\sqrt{2}$ to the power of $1$ .

${(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2}$

Step 1.3.5.3

Raise $\sqrt{2}$ to the power of $1$ .

${(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2}$

Step 1.3.5.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

${(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2}$

Step 1.3.5.5

Add $1$ and $1$ .

${(\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{2}$

Step 1.3.5.6

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.3.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

${(\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2}$

Step 1.3.5.6.2

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

${(\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2}$

Step 1.3.5.6.3

Combine $\frac{1}{2}$ and $2$ .

${(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2}$

Step 1.3.5.6.4

Cancel the common factor of $2$ .

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Step 1.3.5.6.4.1

Cancel the common factor.

${(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2}$

Step 1.3.5.6.4.2

Rewrite the expression.

${(\frac{2 \sqrt{2}}{2^{1}})}^{2}$

Step 1.3.5.6.5

Evaluate the exponent.

${(\frac{2 \sqrt{2}}{2})}^{2}$

Step 1.3.6

Cancel the common factor of $2$ .

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Step 1.3.6.1

Cancel the common factor.

${(\frac{2 \sqrt{2}}{2})}^{2}$

Step 1.3.6.2

Divide $\sqrt{2}$ by $1$ .

${\sqrt{2}}^{2}$

Step 1.3.7

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

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Step 1.3.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

${(2^{\frac{1}{2}})}^{2}$

Step 1.3.7.2

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

$2^{\frac{1}{2} \cdot 2}$

Step 1.3.7.3

Combine $\frac{1}{2}$ and $2$ .

$2^{\frac{2}{2}}$

Step 1.3.7.4

Cancel the common factor of $2$ .

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Step 1.3.7.4.1

Cancel the common factor.

$2^{\frac{2}{2}}$

Step 1.3.7.4.2

Rewrite the expression.

$2^{1}$

Step 1.3.7.5

Evaluate the exponent.

$2$

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 $2$ and a given point $(- \frac{π}{4}, - 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (- 1) = 2 \cdot (x - (- \frac{π}{4}))$

Step 2.2

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

$y + 1 = 2 \cdot (x + \frac{π}{4})$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $2 \cdot (x + \frac{π}{4})$ .

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Step 2.3.1.1

Rewrite.

$y + 1 = 0 + 0 + 2 \cdot (x + \frac{π}{4})$

Step 2.3.1.2

Simplify by adding zeros.

$y + 1 = 2 \cdot (x + \frac{π}{4})$

Step 2.3.1.3

Apply the distributive property.

$y + 1 = 2 x + 2 \frac{π}{4}$

Step 2.3.1.4

Cancel the common factor of $2$ .

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Step 2.3.1.4.1

Factor $2$ out of $4$ .

$y + 1 = 2 x + 2 \frac{π}{2 (2)}$

Step 2.3.1.4.2

Cancel the common factor.

$y + 1 = 2 x + 2 \frac{π}{2 \cdot 2}$

Step 2.3.1.4.3

Rewrite the expression.

$y + 1 = 2 x + \frac{π}{2}$

Step 2.3.2

Subtract $1$ from both sides of the equation.

$y = 2 x + \frac{π}{2} - 1$

Step 2.3.3

Write in $y = m x + b$ form.

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Step 2.3.3.1

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$y = 2 x + \frac{π}{2} - 1 \cdot \frac{2}{2}$

Step 2.3.3.2

Combine $- 1$ and $\frac{2}{2}$ .

$y = 2 x + \frac{π}{2} + \frac{- 1 \cdot 2}{2}$

Step 2.3.3.3

Combine the numerators over the common denominator.

$y = 2 x + \frac{π - 1 \cdot 2}{2}$

Step 2.3.3.4

Multiply $- 1$ by $2$ .

$y = 2 x + \frac{π - 2}{2}$

Step 3