Calculus Examples

$f (x) = 2 t e^{r - \frac{1}{\sqrt{t}}}$

Step 1

Simplify each term.

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Step 1.1

Multiply $\frac{1}{\sqrt{t}}$ by $\frac{\sqrt{t}}{\sqrt{t}}$ .

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

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{1}{\sqrt{t}}$ by $\frac{\sqrt{t}}{\sqrt{t}}$ .

$\frac{d}{d x} [2 t e^{r - \frac{\sqrt{t}}{\sqrt{t} \sqrt{t}}}]$

Step 1.2.2

Raise $\sqrt{t}$ to the power of $1$ .

$\frac{d}{d x} [2 t e^{r - \frac{\sqrt{t}}{{\sqrt{t}}^{1} \sqrt{t}}}]$

Step 1.2.3

Raise $\sqrt{t}$ to the power of $1$ .

$\frac{d}{d x} [2 t e^{r - \frac{\sqrt{t}}{{\sqrt{t}}^{1} {\sqrt{t}}^{1}}}]$

Step 1.2.4

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

$\frac{d}{d x} [2 t e^{r - \frac{\sqrt{t}}{{\sqrt{t}}^{1 + 1}}}]$

Step 1.2.5

Add $1$ and $1$ .

$\frac{d}{d x} [2 t e^{r - \frac{\sqrt{t}}{{\sqrt{t}}^{2}}}]$

Step 1.2.6

Rewrite ${\sqrt{t}}^{2}$ as $t$ .

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Step 1.2.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{t}$ as $t^{\frac{1}{2}}$ .

$\frac{d}{d x} [2 t e^{r - \frac{\sqrt{t}}{{(t^{\frac{1}{2}})}^{2}}}]$

Step 1.2.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{d}{d x} [2 t e^{r - \frac{\sqrt{t}}{t^{\frac{1}{2} \cdot 2}}}]$

Step 1.2.6.3

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

$\frac{d}{d x} [2 t e^{r - \frac{\sqrt{t}}{t^{\frac{2}{2}}}}]$

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

$\frac{d}{d x} [2 t e^{r - \frac{\sqrt{t}}{t^{\frac{2}{2}}}}]$

Step 1.2.6.4.2

Rewrite the expression.

$\frac{d}{d x} [2 t e^{r - \frac{\sqrt{t}}{t^{1}}}]$

Step 1.2.6.5

Simplify.

$\frac{d}{d x} [2 t e^{r - \frac{\sqrt{t}}{t}}]$

Step 2

Since $2 t e^{r - \frac{\sqrt{t}}{t}}$ is constant with respect to $x$ , the derivative of $2 t e^{r - \frac{\sqrt{t}}{t}}$ with respect to $x$ is $0$ .

$0$