Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

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

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

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.3

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

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

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

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

Step 1.3.4

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

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

Step 1.3.5

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

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

Step 1.4

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

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Step 1.4.1

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

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

Step 1.4.2

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

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

Step 1.5

Simplify.

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Step 1.5.1

Apply the distributive property.

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

Step 1.5.2

Combine terms.

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Step 1.5.2.1

Subtract $2 x e^{x}$ from $e^{x} (2 x)$ .

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Step 1.5.2.1.1

Move $e^{x}$ .

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

Step 1.5.2.1.2

Subtract $2 x e^{x}$ from $2 x e^{x}$ .

$x^{2} e^{x} + 0 - 2 e^{x} + 2 e^{x}$

Step 1.5.2.2

Add $x^{2} e^{x}$ and $0$ .

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

Step 1.5.2.3

Add $- 2 e^{x}$ and $2 e^{x}$ .

$x^{2} e^{x} + 0$

Step 1.5.2.4

Add $x^{2} e^{x}$ and $0$ .

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

Step 1.5.3

Reorder the factors of $x^{2} e^{x}$ .

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

Step 1.5.4

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

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

Step 1.6

Evaluate the derivative at $x = 1$ .

${(1)}^{2} e^{1}$

Step 1.7

Simplify.

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Step 1.7.1

One to any power is one.

$1 e^{1}$

Step 1.7.2

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

$e^{1}$

Step 1.7.3

Simplify.

$e$

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

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

Step 2.2

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

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

Step 2.3

Solve for $y$ .

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Step 2.3.1

Simplify $e \cdot (x - 1)$ .

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Step 2.3.1.1

Rewrite.

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

Step 2.3.1.2

Simplify by multiplying through.

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Step 2.3.1.2.1

Apply the distributive property.

$y - e = e x + e \cdot - 1$

Step 2.3.1.2.2

Move $- 1$ to the left of $e$ .

$y - e = e x - 1 \cdot e$

Step 2.3.1.3

Rewrite $- 1 e$ as $- e$ .

$y - e = e x - e$

Step 2.3.2

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

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Step 2.3.2.1

Add $e$ to both sides of the equation.

$y = e x - e + e$

Step 2.3.2.2

Combine the opposite terms in $e x - e + e$ .

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Step 2.3.2.2.1

Add $- e$ and $e$ .

$y = e x + 0$

Step 2.3.2.2.2

Add $e x$ and $0$ .

$y = e x$

Step 3