Calculus Examples

$y = x^{2} - 2^{x}$ , $(3, 1)$

Step 1

Find the first derivative and evaluate at $x = 3$ and $y = 1$ to find the slope of the tangent line.

Tap for more steps...

Step 1.1

Differentiate.

Tap for more steps...

Step 1.1.1

By the Sum Rule, the derivative of $x^{2} - 2^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 2^{x}]$ .

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

Step 1.1.2

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

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

Step 1.2

Evaluate $\frac{d}{d x} [- 2^{x}]$ .

Tap for more steps...

Step 1.2.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- 2^{x}$ with respect to $x$ is $- \frac{d}{d x} [2^{x}]$ .

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

Step 1.2.2

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $2$ .

$2 x - 2^{x} \ln (2)$

Step 1.3

Reorder terms.

$- 2^{x} \ln (2) + 2 x$

Step 1.4

Evaluate the derivative at $x = 3$ .

$- 2^{3} \ln (2) + 2 (3)$

Step 1.5

Simplify each term.

Tap for more steps...

Step 1.5.1

Raise $2$ to the power of $3$ .

$- 1 \cdot 8 \ln (2) + 2 (3)$

Step 1.5.2

Multiply $- 1$ by $8$ .

$- 8 \ln (2) + 2 (3)$

Step 1.5.3

Simplify $- 8 \ln (2)$ by moving $8$ inside the logarithm.

$- \ln (2^{8}) + 2 (3)$

Step 1.5.4

Raise $2$ to the power of $8$ .

$- \ln (256) + 2 (3)$

Step 1.5.5

Multiply $2$ by $3$ .

$- \ln (256) + 6$

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 $- \ln (256) + 6$ and a given point $(3, 1)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (1) = (- \ln (256) + 6) \cdot (x - (3))$

Step 2.2

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

$y - 1 = (- \ln (256) + 6) \cdot (x - 3)$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Simplify $(- \ln (256) + 6) \cdot (x - 3)$ .

Tap for more steps...

Step 2.3.1.1

Rewrite.

$y - 1 = 0 + 0 + (- \ln (256) + 6) \cdot (x - 3)$

Step 2.3.1.2

Simplify by adding zeros.

$y - 1 = (- \ln (256) + 6) \cdot (x - 3)$

Step 2.3.1.3

Expand $(- \ln (256) + 6) (x - 3)$ using the FOIL Method.

Tap for more steps...

Step 2.3.1.3.1

Apply the distributive property.

$y - 1 = - \ln (256) (x - 3) + 6 (x - 3)$

Step 2.3.1.3.2

Apply the distributive property.

$y - 1 = - \ln (256) x - \ln (256) \cdot - 3 + 6 (x - 3)$

Step 2.3.1.3.3

Apply the distributive property.

$y - 1 = - \ln (256) x - \ln (256) \cdot - 3 + 6 x + 6 \cdot - 3$

Step 2.3.1.4

Simplify terms.

Tap for more steps...

Step 2.3.1.4.1

Simplify each term.

Tap for more steps...

Step 2.3.1.4.1.1

Multiply $- \ln (256) \cdot - 3$ .

Tap for more steps...

Step 2.3.1.4.1.1.1

Multiply $- 3$ by $- 1$ .

$y - 1 = - \ln (256) x + 3 \ln (256) + 6 x + 6 \cdot - 3$

Step 2.3.1.4.1.1.2

Simplify $3 \ln (256)$ by moving $3$ inside the logarithm.

$y - 1 = - \ln (256) x + \ln (256^{3}) + 6 x + 6 \cdot - 3$

Step 2.3.1.4.1.2

Raise $256$ to the power of $3$ .

$y - 1 = - \ln (256) x + \ln (16777216) + 6 x + 6 \cdot - 3$

Step 2.3.1.4.1.3

Multiply $6$ by $- 3$ .

$y - 1 = - \ln (256) x + \ln (16777216) + 6 x - 18$

Step 2.3.1.4.2

Reorder factors in $- \ln (256) x + \ln (16777216) + 6 x - 18$ .

$y - 1 = - x \ln (256) + \ln (16777216) + 6 x - 18$

Step 2.3.2

Move all the terms containing a logarithm to the left side of the equation.

$x \ln (256) - \ln (16777216) = - y + 1 + 6 x - 18$

Step 2.3.3

Subtract $18$ from $1$ .

$x \ln (256) - \ln (16777216) = - y + 6 x - 17$

Step 2.3.4

Rewrite the equation as $- y + 6 x - 17 = x \ln (256) - \ln (16777216)$ .

$- y + 6 x - 17 = x \ln (256) - \ln (16777216)$

Step 2.3.5

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

Tap for more steps...

Step 2.3.5.1

Subtract $6 x$ from both sides of the equation.

$- y - 17 = x \ln (256) - \ln (16777216) - 6 x$

Step 2.3.5.2

Add $17$ to both sides of the equation.

$- y = x \ln (256) - \ln (16777216) - 6 x + 17$

Step 2.3.6

Divide each term in $- y = x \ln (256) - \ln (16777216) - 6 x + 17$ by $- 1$ and simplify.

Tap for more steps...

Step 2.3.6.1

Divide each term in $- y = x \ln (256) - \ln (16777216) - 6 x + 17$ by $- 1$ .

$\frac{- y}{- 1} = \frac{x \ln (256)}{- 1} + \frac{- \ln (16777216)}{- 1} + \frac{- 6 x}{- 1} + \frac{17}{- 1}$

Step 2.3.6.2

Simplify the left side.

Tap for more steps...

Step 2.3.6.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{x \ln (256)}{- 1} + \frac{- \ln (16777216)}{- 1} + \frac{- 6 x}{- 1} + \frac{17}{- 1}$

Step 2.3.6.2.2

Divide $y$ by $1$ .

$y = \frac{x \ln (256)}{- 1} + \frac{- \ln (16777216)}{- 1} + \frac{- 6 x}{- 1} + \frac{17}{- 1}$

Step 2.3.6.3

Simplify the right side.

Tap for more steps...

Step 2.3.6.3.1

Simplify each term.

Tap for more steps...

Step 2.3.6.3.1.1

Move the negative one from the denominator of $\frac{x \ln (256)}{- 1}$ .

$y = - 1 \cdot (x \ln (256)) + \frac{- \ln (16777216)}{- 1} + \frac{- 6 x}{- 1} + \frac{17}{- 1}$

Step 2.3.6.3.1.2

Rewrite $- 1 \cdot (x \ln (256))$ as $- (x \ln (256))$ .

$y = - (x \ln (256)) + \frac{- \ln (16777216)}{- 1} + \frac{- 6 x}{- 1} + \frac{17}{- 1}$

Step 2.3.6.3.1.3

Dividing two negative values results in a positive value.

$y = - x \ln (256) + \frac{\ln (16777216)}{1} + \frac{- 6 x}{- 1} + \frac{17}{- 1}$

Step 2.3.6.3.1.4

Divide $\ln (16777216)$ by $1$ .

$y = - x \ln (256) + \ln (16777216) + \frac{- 6 x}{- 1} + \frac{17}{- 1}$

Step 2.3.6.3.1.5

Move the negative one from the denominator of $\frac{- 6 x}{- 1}$ .

$y = - x \ln (256) + \ln (16777216) - 1 \cdot (- 6 x) + \frac{17}{- 1}$

Step 2.3.6.3.1.6

Rewrite $- 1 \cdot (- 6 x)$ as $- (- 6 x)$ .

$y = - x \ln (256) + \ln (16777216) - (- 6 x) + \frac{17}{- 1}$

Step 2.3.6.3.1.7

Multiply $- 6$ by $- 1$ .

$y = - x \ln (256) + \ln (16777216) + 6 x + \frac{17}{- 1}$

Step 2.3.6.3.1.8

Divide $17$ by $- 1$ .

$y = - x \ln (256) + \ln (16777216) + 6 x - 17$

Step 2.3.7

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

Tap for more steps...

Step 2.3.7.1

Move $\ln (16777216)$ .

$y = - x \ln (256) + 6 x + \ln (16777216) - 17$

Step 2.3.7.2

Factor $x$ out of $- x \ln (256)$ .

$y = x (- 1 \ln (256)) + 6 x + \ln (16777216) - 17$

Step 2.3.7.3

Factor $x$ out of $6 x$ .

$y = x (- 1 \ln (256)) + x \cdot 6 + \ln (16777216) - 17$

Step 2.3.7.4

Factor $x$ out of $x (- 1 \ln (256)) + x \cdot 6$ .

$y = x (- 1 \ln (256) + 6) + \ln (16777216) - 17$

Step 2.3.7.5

Reorder $x$ and $- 1 \ln (256) + 6$ .

$y = (- 1 \ln (256) + 6) x + \ln (16777216) - 17$

Step 3