Calculus Examples

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

Step 1

Find the first derivative and evaluate at $x = 0$ and $y = 0$ to find the slope of the tangent line.

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Step 1.1

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

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

Step 1.2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x$ and $g (x) = e^{x}$ .

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

Step 1.3

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

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

Step 1.4

Differentiate using the Power Rule.

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Step 1.4.1

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

$2 (x e^{x} + e^{x} \cdot 1)$

Step 1.4.2

Multiply $e^{x}$ by $1$ .

$2 (x e^{x} + e^{x})$

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

$2 (x e^{x}) + 2 e^{x}$

Step 1.5.2

Reorder terms.

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

Step 1.5.3

Reorder factors in $2 e^{x} x + 2 e^{x}$ .

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

Step 1.6

Evaluate the derivative at $x = 0$ .

$2 (0) \cdot e^{0} + 2 e^{0}$

Step 1.7

Simplify.

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Step 1.7.1

Simplify each term.

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Step 1.7.1.1

Multiply $2$ by $0$ .

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

Step 1.7.1.2

Anything raised to $0$ is $1$ .

$0 \cdot 1 + 2 e^{0}$

Step 1.7.1.3

Multiply $0$ by $1$ .

$0 + 2 e^{0}$

Step 1.7.1.4

Anything raised to $0$ is $1$ .

$0 + 2 \cdot 1$

Step 1.7.1.5

Multiply $2$ by $1$ .

$0 + 2$

Step 1.7.2

Add $0$ and $2$ .

$2$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

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

Step 2.3.2

Add $x$ and $0$ .

$y = 2 x$

Step 3