Calculus Examples

$y = (r^{2} - 2 r) e^{r}$

Step 1

Differentiate using the Product Rule which states that $\frac{d}{d r} [f (r) g (r)]$ is $f (r) \frac{d}{d r} [g (r)] + g (r) \frac{d}{d r} [f (r)]$ where $f (r) = r^{2} - 2 r$ and $g (r) = e^{r}$ .

$(r^{2} - 2 r) \frac{d}{d r} [e^{r}] + e^{r} \frac{d}{d r} [r^{2} - 2 r]$

Step 2

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

$(r^{2} - 2 r) e^{r} + e^{r} \frac{d}{d r} [r^{2} - 2 r]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $r^{2} - 2 r$ with respect to $r$ is $\frac{d}{d r} [r^{2}] + \frac{d}{d r} [- 2 r]$ .

$(r^{2} - 2 r) e^{r} + e^{r} (\frac{d}{d r} [r^{2}] + \frac{d}{d r} [- 2 r])$

Step 3.2

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

$(r^{2} - 2 r) e^{r} + e^{r} (2 r + \frac{d}{d r} [- 2 r])$

Step 3.3

Since $- 2$ is constant with respect to $r$ , the derivative of $- 2 r$ with respect to $r$ is $- 2 \frac{d}{d r} [r]$ .

$(r^{2} - 2 r) e^{r} + e^{r} (2 r - 2 \frac{d}{d r} [r])$

Step 3.4

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

$(r^{2} - 2 r) e^{r} + e^{r} (2 r - 2 \cdot 1)$

Step 3.5

Multiply $- 2$ by $1$ .

$(r^{2} - 2 r) e^{r} + e^{r} (2 r - 2)$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$r^{2} e^{r} - 2 r e^{r} + e^{r} (2 r - 2)$

Step 4.2

Apply the distributive property.

$r^{2} e^{r} - 2 r e^{r} + e^{r} (2 r) + e^{r} \cdot - 2$

Step 4.3

Combine terms.

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Step 4.3.1

Move $- 2$ to the left of $e^{r}$ .

$r^{2} e^{r} - 2 r e^{r} + e^{r} (2 r) - 2 \cdot e^{r}$

Step 4.3.2

Add $- 2 r e^{r}$ and $e^{r} (2 r)$ .

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Step 4.3.2.1

Move $e^{r}$ .

$r^{2} e^{r} - 2 r e^{r} + 2 r e^{r} - 2 e^{r}$

Step 4.3.2.2

Add $- 2 r e^{r}$ and $2 r e^{r}$ .

$r^{2} e^{r} + 0 - 2 e^{r}$

Step 4.3.3

Add $r^{2} e^{r}$ and $0$ .

$r^{2} e^{r} - 2 e^{r}$

Step 4.4

Reorder terms.

$e^{r} r^{2} - 2 e^{r}$

Step 4.5

Reorder factors in $e^{r} r^{2} - 2 e^{r}$ .

$r^{2} e^{r} - 2 e^{r}$