Calculus Examples

$y = x^{4} + 2 e^{x}$ , $(0, 2)$

Step 1

Find the first derivative and evaluate at $x = 0$ and $y = 2$ 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 $x^{4} + 2 e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [2 e^{x}]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [2 e^{x}]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [2 e^{x}]$

Step 1.2

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

Tap for more steps...

Step 1.2.1

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

$4 x^{3} + 2 \frac{d}{d x} [e^{x}]$

Step 1.2.2

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

$4 x^{3} + 2 e^{x}$

Step 1.3

Evaluate the derivative at $x = 0$ .

$4 {(0)}^{3} + 2 e^{0}$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Simplify each term.

Tap for more steps...

Step 1.4.1.1

Raising $0$ to any positive power yields $0$ .

$4 \cdot 0 + 2 e^{0}$

Step 1.4.1.2

Multiply $4$ by $0$ .

$0 + 2 e^{0}$

Step 1.4.1.3

Anything raised to $0$ is $1$ .

$0 + 2 \cdot 1$

Step 1.4.1.4

Multiply $2$ by $1$ .

$0 + 2$

Step 1.4.2

Add $0$ and $2$ .

$2$

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 $2$ and a given point $(0, 2)$ 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 $m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ .

$y - (2) = 2 \cdot (x - (0))$

Step 2.2

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

$y - 2 = 2 \cdot (x + 0)$

Step 2.3

Solve for $y$ .

Tap for more steps...

Step 2.3.1

Add $x$ and $0$ .

$y - 2 = 2 x$

Step 2.3.2

Add $2$ to both sides of the equation.

$y = 2 x + 2$

Step 3