Calculus Examples

$y = 8 x e^{x}$ , $(0, 0)$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (8 x e^{x})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

Since $8$ is constant with respect to $x$ , the derivative of $8 x e^{x}$ with respect to $x$ is $8 \frac{d}{d x} [x e^{x}]$ .

$8 \frac{d}{d x} [x e^{x}]$

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

$8 (x \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x])$

Step 3.3

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

$8 (x e^{x} + e^{x} \frac{d}{d x} [x])$

Step 3.4

Differentiate using the Power Rule.

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Step 3.4.1

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

$8 (x e^{x} + e^{x} \cdot 1)$

Step 3.4.2

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

$8 (x e^{x} + e^{x})$

Step 3.5

Simplify.

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Step 3.5.1

Apply the distributive property.

$8 (x e^{x}) + 8 e^{x}$

Step 3.5.2

Reorder terms.

$8 e^{x} x + 8 e^{x}$

Step 3.5.3

Reorder factors in $8 e^{x} x + 8 e^{x}$ .

$8 x e^{x} + 8 e^{x}$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = 8 x e^{x} + 8 e^{x}$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = 8 x e^{x} + 8 e^{x}$

Step 6

Replace $x$ with $0$ and $y$ with $0$ in the expression.

$\frac{d y}{d x} = 8 (0) e^{0} + 8 e^{0}$

Step 7

Simplify the result.

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Step 7.1

Simplify each term.

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Step 7.1.1

Multiply $8$ by $0$ .

$0 e^{0} + 8 e^{0}$

Step 7.1.2

Anything raised to $0$ is $1$ .

$0 \cdot 1 + 8 e^{0}$

Step 7.1.3

Multiply $0$ by $1$ .

$0 + 8 e^{0}$

Step 7.1.4

Anything raised to $0$ is $1$ .

$0 + 8 \cdot 1$

Step 7.1.5

Multiply $8$ by $1$ .

$0 + 8$

Step 7.2

Add $0$ and $8$ .

$8$