Calculus Examples

$z = x^{\frac{3}{2}} (x + c e^{x})$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (z) = \frac{d}{d x} (x^{\frac{3}{2}} (x + c e^{x}))$

Step 2

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

$z'$

Step 3

Differentiate the right side of the equation.

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Step 3.1

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 $f (x) = x^{\frac{3}{2}}$ and $g (x) = x + c e^{x}$ .

$x^{\frac{3}{2}} \frac{d}{d x} [x + c e^{x}] + (x + c e^{x}) \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 3.2

Differentiate.

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Step 3.2.1

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

$x^{\frac{3}{2}} (\frac{d}{d x} [x] + \frac{d}{d x} [c e^{x}]) + (x + c e^{x}) \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 3.2.2

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

$x^{\frac{3}{2}} (1 + \frac{d}{d x} [c e^{x}]) + (x + c e^{x}) \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 3.2.3

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

$x^{\frac{3}{2}} (1 + c \frac{d}{d x} [e^{x}]) + (x + c e^{x}) \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 3.3

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

$x^{\frac{3}{2}} (1 + c e^{x}) + (x + c e^{x}) \frac{d}{d x} [x^{\frac{3}{2}}]$

Step 3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = \frac{3}{2}$ .

$x^{\frac{3}{2}} (1 + c e^{x}) + (x + c e^{x}) (\frac{3}{2} x^{\frac{3}{2} - 1})$

Step 3.5

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{\frac{3}{2}} (1 + c e^{x}) + (x + c e^{x}) (\frac{3}{2} x^{\frac{3}{2} - 1 \cdot \frac{2}{2}})$

Step 3.6

Combine $- 1$ and $\frac{2}{2}$ .

$x^{\frac{3}{2}} (1 + c e^{x}) + (x + c e^{x}) (\frac{3}{2} x^{\frac{3}{2} + \frac{- 1 \cdot 2}{2}})$

Step 3.7

Combine the numerators over the common denominator.

$x^{\frac{3}{2}} (1 + c e^{x}) + (x + c e^{x}) (\frac{3}{2} x^{\frac{3 - 1 \cdot 2}{2}})$

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply $- 1$ by $2$ .

$x^{\frac{3}{2}} (1 + c e^{x}) + (x + c e^{x}) (\frac{3}{2} x^{\frac{3 - 2}{2}})$

Step 3.8.2

Subtract $2$ from $3$ .

$x^{\frac{3}{2}} (1 + c e^{x}) + (x + c e^{x}) (\frac{3}{2} x^{\frac{1}{2}})$

Step 3.9

Combine $\frac{3}{2}$ and $x^{\frac{1}{2}}$ .

$x^{\frac{3}{2}} (1 + c e^{x}) + (x + c e^{x}) \frac{3 x^{\frac{1}{2}}}{2}$

Step 3.10

Simplify.

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Step 3.10.1

Apply the distributive property.

$x^{\frac{3}{2}} \cdot 1 + x^{\frac{3}{2}} (c e^{x}) + (x + c e^{x}) \frac{3 x^{\frac{1}{2}}}{2}$

Step 3.10.2

Apply the distributive property.

$x^{\frac{3}{2}} \cdot 1 + x^{\frac{3}{2}} (c e^{x}) + x \frac{3 x^{\frac{1}{2}}}{2} + c e^{x} \frac{3 x^{\frac{1}{2}}}{2}$

Step 3.10.3

Combine terms.

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Step 3.10.3.1

Multiply $x^{\frac{3}{2}}$ by $1$ .

$x^{\frac{3}{2}} + x^{\frac{3}{2}} (c e^{x}) + x \frac{3 x^{\frac{1}{2}}}{2} + c e^{x} \frac{3 x^{\frac{1}{2}}}{2}$

Step 3.10.3.2

Combine $x$ and $\frac{3 x^{\frac{1}{2}}}{2}$ .

$x^{\frac{3}{2}} + x^{\frac{3}{2}} (c e^{x}) + \frac{x (3 x^{\frac{1}{2}})}{2} + c e^{x} \frac{3 x^{\frac{1}{2}}}{2}$

Step 3.10.3.3

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

$x^{\frac{3}{2}} + x^{\frac{3}{2}} (c e^{x}) + \frac{3 (x^{1} x^{\frac{1}{2}})}{2} + c e^{x} \frac{3 x^{\frac{1}{2}}}{2}$

Step 3.10.3.4

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

$x^{\frac{3}{2}} + x^{\frac{3}{2}} (c e^{x}) + \frac{3 x^{1 + \frac{1}{2}}}{2} + c e^{x} \frac{3 x^{\frac{1}{2}}}{2}$

Step 3.10.3.5

Write $1$ as a fraction with a common denominator.

$x^{\frac{3}{2}} + x^{\frac{3}{2}} (c e^{x}) + \frac{3 x^{\frac{2}{2} + \frac{1}{2}}}{2} + c e^{x} \frac{3 x^{\frac{1}{2}}}{2}$

Step 3.10.3.6

Combine the numerators over the common denominator.

$x^{\frac{3}{2}} + x^{\frac{3}{2}} (c e^{x}) + \frac{3 x^{\frac{2 + 1}{2}}}{2} + c e^{x} \frac{3 x^{\frac{1}{2}}}{2}$

Step 3.10.3.7

Add $2$ and $1$ .

$x^{\frac{3}{2}} + x^{\frac{3}{2}} (c e^{x}) + \frac{3 x^{\frac{3}{2}}}{2} + c e^{x} \frac{3 x^{\frac{1}{2}}}{2}$

Step 3.10.3.8

Combine $\frac{3 x^{\frac{1}{2}}}{2}$ and $c$ .

$x^{\frac{3}{2}} + x^{\frac{3}{2}} (c e^{x}) + \frac{3 x^{\frac{3}{2}}}{2} + \frac{3 x^{\frac{1}{2}} c}{2} e^{x}$

Step 3.10.3.9

Combine $\frac{3 x^{\frac{1}{2}} c}{2}$ and $e^{x}$ .

$x^{\frac{3}{2}} + x^{\frac{3}{2}} (c e^{x}) + \frac{3 x^{\frac{3}{2}}}{2} + \frac{3 x^{\frac{1}{2}} c e^{x}}{2}$

Step 3.10.3.10

To write $x^{\frac{3}{2}}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$x^{\frac{3}{2}} (c e^{x}) + x^{\frac{3}{2}} \cdot \frac{2}{2} + \frac{3 x^{\frac{3}{2}}}{2} + \frac{3 x^{\frac{1}{2}} c e^{x}}{2}$

Step 3.10.3.11

Combine $x^{\frac{3}{2}}$ and $\frac{2}{2}$ .

$x^{\frac{3}{2}} (c e^{x}) + \frac{x^{\frac{3}{2}} \cdot 2}{2} + \frac{3 x^{\frac{3}{2}}}{2} + \frac{3 x^{\frac{1}{2}} c e^{x}}{2}$

Step 3.10.3.12

Combine the numerators over the common denominator.

$x^{\frac{3}{2}} (c e^{x}) + \frac{x^{\frac{3}{2}} \cdot 2 + 3 x^{\frac{3}{2}}}{2} + \frac{3 x^{\frac{1}{2}} c e^{x}}{2}$

Step 3.10.3.13

Move $2$ to the left of $x^{\frac{3}{2}}$ .

$x^{\frac{3}{2}} (c e^{x}) + \frac{2 \cdot x^{\frac{3}{2}} + 3 x^{\frac{3}{2}}}{2} + \frac{3 x^{\frac{1}{2}} c e^{x}}{2}$

Step 3.10.3.14

Add $2 x^{\frac{3}{2}}$ and $3 x^{\frac{3}{2}}$ .

$x^{\frac{3}{2}} (c e^{x}) + \frac{5 x^{\frac{3}{2}}}{2} + \frac{3 x^{\frac{1}{2}} c e^{x}}{2}$

Step 3.10.4

Reorder terms.

$e^{x} x^{\frac{3}{2}} c + \frac{3 x^{\frac{1}{2}} c e^{x}}{2} + \frac{5 x^{\frac{3}{2}}}{2}$

Step 4

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

$z' = e^{x} x^{\frac{3}{2}} c + \frac{3 x^{\frac{1}{2}} c e^{x}}{2} + \frac{5 x^{\frac{3}{2}}}{2}$

Step 5

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

$\frac{d z}{d x} = e^{x} x^{\frac{3}{2}} c + \frac{3 x^{\frac{1}{2}} c e^{x}}{2} + \frac{5 x^{\frac{3}{2}}}{2}$