Algebra Examples

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

Step 1

Since $x^{\frac{3}{2}}$ is constant with respect to $c$ , the derivative of $x^{\frac{3}{2}} (x + c e^{x})$ with respect to $c$ is $x^{\frac{3}{2}} \frac{d}{d c} [x + c e^{x}]$ .

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

Step 2

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

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

Step 3

Since $x$ is constant with respect to $c$ , the derivative of $x$ with respect to $c$ is $0$ .

$x^{\frac{3}{2}} (0 + \frac{d}{d c} [c e^{x}])$

Step 4

Add $0$ and $\frac{d}{d c} [c e^{x}]$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify the expression.

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Step 7.1

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

$x^{\frac{3}{2}} e^{x}$

Step 7.2

Reorder the factors of $x^{\frac{3}{2}} e^{x}$ .

$e^{x} x^{\frac{3}{2}}$