Calculus Examples

$\tan^{2} (x)$ , $(\frac{π}{4}, 1)$

Step 1

Write $\tan^{2} (x)$ as an equation.

$y = \tan^{2} (x)$

Step 2

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

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To apply the Chain Rule, set $u$ as $\tan (x)$ .

$\frac{d}{d u} [u^{2}] \frac{d}{d x} [\tan (x)]$

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

$2 u \frac{d}{d x} [\tan (x)]$

Replace all occurrences of $u$ with $\tan (x)$ .

$2 \tan (x) \frac{d}{d x} [\tan (x)]$

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

$2 \tan (x) \sec^{2} (x)$

Reorder the factors of $2 \tan (x) \sec^{2} (x)$ .

$2 \sec^{2} (x) \tan (x)$

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

$2 \sec^{2} (\frac{π}{4}) \tan (\frac{π}{4})$

Simplify.

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

$2 {(\frac{2}{\sqrt{2}})}^{2} \tan (\frac{π}{4})$

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

$2 {(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})}^{2} \tan (\frac{π}{4})$

Combine and simplify the denominator.

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

$2 {(\frac{2 \sqrt{2}}{\sqrt{2} \sqrt{2}})}^{2} \tan (\frac{π}{4})$

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

$2 {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})}^{2} \tan (\frac{π}{4})$

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

$2 {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})}^{2} \tan (\frac{π}{4})$

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

$2 {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{1 + 1}})}^{2} \tan (\frac{π}{4})$

Add $1$ and $1$ .

$2 {(\frac{2 \sqrt{2}}{{\sqrt{2}}^{2}})}^{2} \tan (\frac{π}{4})$

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

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

$2 {(\frac{2 \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})}^{2} \tan (\frac{π}{4})$

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

$2 {(\frac{2 \sqrt{2}}{2^{\frac{1}{2} \cdot 2}})}^{2} \tan (\frac{π}{4})$

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

$2 {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2} \tan (\frac{π}{4})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$2 {(\frac{2 \sqrt{2}}{2^{\frac{2}{2}}})}^{2} \tan (\frac{π}{4})$

Rewrite the expression.

$2 {(\frac{2 \sqrt{2}}{2^{1}})}^{2} \tan (\frac{π}{4})$

Evaluate the exponent.

$2 {(\frac{2 \sqrt{2}}{2})}^{2} \tan (\frac{π}{4})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$2 {(\frac{2 \sqrt{2}}{2})}^{2} \tan (\frac{π}{4})$

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

$2 {\sqrt{2}}^{2} \tan (\frac{π}{4})$

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

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

$2 {(2^{\frac{1}{2}})}^{2} \tan (\frac{π}{4})$

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

$2 \cdot 2^{\frac{1}{2} \cdot 2} \tan (\frac{π}{4})$

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

$2 \cdot 2^{\frac{2}{2}} \tan (\frac{π}{4})$

Cancel the common factor of $2$ .

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Cancel the common factor.

$2 \cdot 2^{\frac{2}{2}} \tan (\frac{π}{4})$

Rewrite the expression.

$2 \cdot 2^{1} \tan (\frac{π}{4})$

Evaluate the exponent.

$2 \cdot 2 \tan (\frac{π}{4})$

Multiply $2$ by $2$ .

$4 \tan (\frac{π}{4})$

The exact value of $\tan (\frac{π}{4})$ is $1$ .

$4 \cdot 1$

Multiply $4$ by $1$ .

$4$

Step 3

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

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Use the slope $4$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

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

$y - 1 = 4 \cdot (x - \frac{π}{4})$

Solve for $y$ .

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

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Rewrite.

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

Simplify by adding zeros.

$y - 1 = 4 \cdot (x - \frac{π}{4})$

Apply the distributive property.

$y - 1 = 4 x + 4 (- \frac{π}{4})$

Cancel the common factor of $4$ .

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Move the leading negative in $- \frac{π}{4}$ into the numerator.

$y - 1 = 4 x + 4 \frac{- π}{4}$

Cancel the common factor.

$y - 1 = 4 x + 4 \frac{- π}{4}$

Rewrite the expression.

$y - 1 = 4 x - π$

Add $1$ to both sides of the equation.

$y = 4 x - π + 1$

Step 4