Calculus Examples

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

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

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 $g (x) = e^{- x^{2}}$ .

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

Step 1.2

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

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Step 1.2.1

To apply the Chain Rule, set $u$ as $- x^{2}$ .

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

Step 1.2.2

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

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

Step 1.2.3

Replace all occurrences of $u$ with $- x^{2}$ .

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

Step 1.3

Differentiate.

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Multiply $2$ by $- 1$ .

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

Step 1.4

Raise $x$ to the power of $1$ .

$x^{1} x (e^{- x^{2}} \cdot (- 2)) + e^{- x^{2}} \frac{d}{d x} [x]$

Step 1.5

Raise $x$ to the power of $1$ .

$x^{1} x^{1} (e^{- x^{2}} \cdot (- 2)) + e^{- x^{2}} \frac{d}{d x} [x]$

Step 1.6

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

$x^{1 + 1} (e^{- x^{2}} \cdot (- 2)) + e^{- x^{2}} \frac{d}{d x} [x]$

Step 1.7

Simplify the expression.

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Step 1.7.1

Add $1$ and $1$ .

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

Step 1.7.2

Move $- 2$ to the left of $e^{- x^{2}}$ .

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

Step 1.8

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

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

Step 1.9

Multiply $e^{- x^{2}}$ by $1$ .

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

Step 1.10

Simplify.

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Step 1.10.1

Reorder terms.

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

Step 1.10.2

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

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

Step 1.11

Evaluate the derivative at $x = 0$ .

$- 2 {(0)}^{2} e^{- {(0)}^{2}} + e^{- {(0)}^{2}}$

Step 1.12

Simplify.

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Step 1.12.1

Simplify each term.

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Step 1.12.1.1

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

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

Step 1.12.1.2

Multiply $- 2$ by $0$ .

$0 e^{- {(0)}^{2}} + e^{- {(0)}^{2}}$

Step 1.12.1.3

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

$0 e^{- 0} + e^{- {(0)}^{2}}$

Step 1.12.1.4

Multiply $- 1$ by $0$ .

$0 e^{0} + e^{- {(0)}^{2}}$

Step 1.12.1.5

Anything raised to $0$ is $1$ .

$0 \cdot 1 + e^{- {(0)}^{2}}$

Step 1.12.1.6

Multiply $0$ by $1$ .

$0 + e^{- {(0)}^{2}}$

Step 1.12.1.7

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

$0 + e^{- 0}$

Step 1.12.1.8

Multiply $- 1$ by $0$ .

$0 + e^{0}$

Step 1.12.1.9

Anything raised to $0$ is $1$ .

$0 + 1$

Step 1.12.2

Add $0$ and $1$ .

$1$

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

$y - (0) = 1 \cdot (x - (0))$

Step 2.2

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

$y + 0 = 1 \cdot (x + 0)$

Step 2.3

Solve for $y$ .

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Step 2.3.1

Add $y$ and $0$ .

$y = 1 \cdot (x + 0)$

Step 2.3.2

Simplify $1 \cdot (x + 0)$ .

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Step 2.3.2.1

Multiply $x + 0$ by $1$ .

$y = x + 0$

Step 2.3.2.2

Add $x$ and $0$ .

$y = x$

Step 3