Algebra Examples

$7.8 e^{\frac{x}{3} \ln (5)}$

Step 1

Differentiate using the Constant Multiple Rule.

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Step 1.1

Combine $\frac{x}{3}$ and $\ln (5)$ .

$\frac{d}{d x} [7.8 e^{\frac{x \ln (5)}{3}}]$

Step 1.2

Since $7.8$ is constant with respect to $x$ , the derivative of $7.8 e^{\frac{x \ln (5)}{3}}$ with respect to $x$ is $7.8 \frac{d}{d x} [e^{\frac{x \ln (5)}{3}}]$ .

$7.8 \frac{d}{d x} [e^{\frac{x \ln (5)}{3}}]$

Step 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 $g (x) = \frac{x \ln (5)}{3}$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $\frac{x \ln (5)}{3}$ .

$7.8 (\frac{d}{d u} [e^{u}] \frac{d}{d x} [\frac{x \ln (5)}{3}])$

Step 2.2

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

$7.8 (e^{u} \frac{d}{d x} [\frac{x \ln (5)}{3}])$

Step 2.3

Replace all occurrences of $u$ with $\frac{x \ln (5)}{3}$ .

$7.8 (e^{\frac{x \ln (5)}{3}} \frac{d}{d x} [\frac{x \ln (5)}{3}])$

Step 3

Differentiate.

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Step 3.1

Since $\frac{\ln (5)}{3}$ is constant with respect to $x$ , the derivative of $\frac{x \ln (5)}{3}$ with respect to $x$ is $\frac{\ln (5)}{3} \frac{d}{d x} [x]$ .

$7.8 e^{\frac{x \ln (5)}{3}} (\frac{\ln (5)}{3} \frac{d}{d x} [x])$

Step 3.2

Simplify terms.

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Step 3.2.1

Combine $\frac{\ln (5)}{3}$ and $7.8$ .

$\frac{\ln (5) \cdot 7.8}{3} e^{\frac{x \ln (5)}{3}} \frac{d}{d x} [x]$

Step 3.2.2

Multiply $\ln (5)$ by $7.8$ .

$\frac{12.55361571}{3} e^{\frac{x \ln (5)}{3}} \frac{d}{d x} [x]$

Step 3.2.3

Combine $\frac{12.55361571}{3}$ and $e^{\frac{x \ln (5)}{3}}$ .

$\frac{12.55361571 e^{\frac{x \ln (5)}{3}}}{3} \frac{d}{d x} [x]$

Step 3.3

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

$\frac{12.55361571 e^{\frac{x \ln (5)}{3}}}{3} \cdot 1$

Step 3.4

Multiply $\frac{12.55361571 e^{\frac{x \ln (5)}{3}}}{3}$ by $1$ .

$\frac{12.55361571 e^{\frac{x \ln (5)}{3}}}{3}$

Step 4

Simplify.

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Step 4.1

Factor $12.55361571$ out of $12.55361571 e^{\frac{x \ln (5)}{3}}$ .

$\frac{12.55361571 (e^{\frac{x \ln (5)}{3}})}{3}$

Step 4.2

Factor $3$ out of $3$ .

$\frac{12.55361571 (e^{\frac{x \ln (5)}{3}})}{3 (1)}$

Step 4.3

Separate fractions.

$\frac{12.55361571}{3} \cdot \frac{e^{\frac{x \ln (5)}{3}}}{1}$

Step 4.4

Divide $12.55361571$ by $3$ .

$4.18453857 \frac{e^{\frac{x \ln (5)}{3}}}{1}$

Step 4.5

Divide $e^{\frac{x \ln (5)}{3}}$ by $1$ .

$4.18453857 e^{\frac{x \ln (5)}{3}}$