Calculus Examples

$f (x) = \tan^{2} (x)$ , $(- \frac{π}{4}, 1)$

Step 1

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

Tap for more steps...

Step 1.1

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 $g (x) = \tan (x)$ .

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

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

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

Step 1.5

Simplify.

Tap for more steps...

Step 1.5.1

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

$2 \sec^{2} (\frac{7 π}{4}) \tan (- \frac{π}{4})$

Step 1.5.2

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

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

Step 1.5.3

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

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

Step 1.5.4

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

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

Step 1.5.5

Combine and simplify the denominator.

Tap for more steps...

Step 1.5.5.1

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

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

Step 1.5.5.2

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

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

Step 1.5.5.3

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

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

Step 1.5.5.4

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

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

Step 1.5.5.5

Add $1$ and $1$ .

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

Step 1.5.5.6

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

Tap for more steps...

Step 1.5.5.6.1

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

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

Step 1.5.5.6.2

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

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

Step 1.5.5.6.3

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

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

Step 1.5.5.6.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.5.6.4.1

Cancel the common factor.

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

Step 1.5.5.6.4.2

Rewrite the expression.

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

Step 1.5.5.6.5

Evaluate the exponent.

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

Step 1.5.6

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.6.1

Cancel the common factor.

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

Step 1.5.6.2

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

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

Step 1.5.7

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

Tap for more steps...

Step 1.5.7.1

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

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

Step 1.5.7.2

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

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

Step 1.5.7.3

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

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

Step 1.5.7.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.5.7.4.1

Cancel the common factor.

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

Step 1.5.7.4.2

Rewrite the expression.

$2 \cdot 2^{1} \tan (- \frac{π}{4})$

Step 1.5.7.5

Evaluate the exponent.

$2 \cdot 2 \tan (- \frac{π}{4})$

Step 1.5.8

Multiply $2$ by $2$ .

$4 \tan (- \frac{π}{4})$

Step 1.5.9

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

$4 \tan (\frac{7 π}{4})$

Step 1.5.10

Apply the reference angle by finding the angle with equivalent trig values in the first quadrant. Make the expression negative because tangent is negative in the fourth quadrant.

$4 (- \tan (\frac{π}{4}))$

Step 1.5.11

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

$4 (- 1 \cdot 1)$

Step 1.5.12

Multiply $4 (- 1 \cdot 1)$ .

Tap for more steps...

Step 1.5.12.1

Multiply $- 1$ by $1$ .

$4 \cdot - 1$

Step 1.5.12.2

Multiply $4$ by $- 1$ .

$- 4$

Step 2

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

Tap for more steps...

Step 2.1

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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

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

Step 2.2

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

$y - 1 = - 4 \cdot (x + \frac{π}{4})$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

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

Tap for more steps...

Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = - 4 \cdot (x + \frac{π}{4})$

Step 2.3.1.3

Apply the distributive property.

$y - 1 = - 4 x - 4 \frac{π}{4}$

Step 2.3.1.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 2.3.1.4.1

Factor $4$ out of $- 4$ .

$y - 1 = - 4 x + 4 (- 1) \frac{π}{4}$

Step 2.3.1.4.2

Cancel the common factor.

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

Step 2.3.1.4.3

Rewrite the expression.

$y - 1 = - 4 x - 1 π$

Step 2.3.1.5

Rewrite $- 1 π$ as $- π$ .

$y - 1 = - 4 x - π$

Step 2.3.2

Add $1$ to both sides of the equation.

$y = - 4 x - π + 1$

Step 3