Calculus Examples

$(x^{2} - \frac{5}{4}) e^{x}$

Step 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 $f (x) = x^{2} - \frac{5}{4}$ and $g (x) = e^{x}$ .

$(x^{2} - \frac{5}{4}) \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{2} - \frac{5}{4}]$

Step 2

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

$(x^{2} - \frac{5}{4}) e^{x} + e^{x} \frac{d}{d x} [x^{2} - \frac{5}{4}]$

Step 3

Differentiate.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $x^{2} - \frac{5}{4}$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{5}{4}]$ .

$(x^{2} - \frac{5}{4}) e^{x} + e^{x} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- \frac{5}{4}])$

Step 3.2

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

$(x^{2} - \frac{5}{4}) e^{x} + e^{x} (2 x + \frac{d}{d x} [- \frac{5}{4}])$

Step 3.3

Since $- \frac{5}{4}$ is constant with respect to $x$ , the derivative of $- \frac{5}{4}$ with respect to $x$ is $0$ .

$(x^{2} - \frac{5}{4}) e^{x} + e^{x} (2 x + 0)$

Step 3.4

Add $2 x$ and $0$ .

$(x^{2} - \frac{5}{4}) e^{x} + e^{x} (2 x)$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

$x^{2} e^{x} - \frac{5}{4} e^{x} + e^{x} (2 x)$

Step 4.2

Reorder terms.

$e^{x} x^{2} - \frac{5}{4} e^{x} + 2 e^{x} x$

Step 4.3

Simplify each term.

Tap for more steps...

Step 4.3.1

Combine $e^{x}$ and $\frac{5}{4}$ .

$e^{x} x^{2} - \frac{e^{x} \cdot 5}{4} + 2 e^{x} x$

Step 4.3.2

Move $5$ to the left of $e^{x}$ .

$e^{x} x^{2} - \frac{5 e^{x}}{4} + 2 e^{x} x$

Step 4.4

Subtract $\frac{5 e^{x}}{4}$ from $e^{x} x^{2}$ .

Tap for more steps...

Step 4.4.1

Reorder $e^{x}$ and $x^{2}$ .

$x^{2} e^{x} - \frac{5 e^{x}}{4} + 2 e^{x} x$

Step 4.4.2

To write $x^{2} e^{x}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$x^{2} e^{x} \cdot \frac{4}{4} - \frac{5 e^{x}}{4} + 2 e^{x} x$

Step 4.4.3

Combine $x^{2} e^{x}$ and $\frac{4}{4}$ .

$\frac{x^{2} e^{x} \cdot 4}{4} - \frac{5 e^{x}}{4} + 2 e^{x} x$

Step 4.4.4

Combine the numerators over the common denominator.

$\frac{x^{2} e^{x} \cdot 4 - 5 e^{x}}{4} + 2 e^{x} x$

Step 4.5

Simplify the numerator.

Tap for more steps...

Step 4.5.1

Move $4$ to the left of $x^{2} e^{x}$ .

$\frac{4 \cdot (x^{2} e^{x}) - 5 e^{x}}{4} + 2 e^{x} x$

Step 4.5.2

Factor $e^{x}$ out of $4 x^{2} e^{x} - 5 e^{x}$ .

Tap for more steps...

Step 4.5.2.1

Factor $e^{x}$ out of $4 x^{2} e^{x}$ .

$\frac{e^{x} (4 x^{2}) - 5 e^{x}}{4} + 2 e^{x} x$

Step 4.5.2.2

Factor $e^{x}$ out of $- 5 e^{x}$ .

$\frac{e^{x} (4 x^{2}) + e^{x} \cdot - 5}{4} + 2 e^{x} x$

Step 4.5.2.3

Factor $e^{x}$ out of $e^{x} (4 x^{2}) + e^{x} \cdot - 5$ .

$\frac{e^{x} (4 x^{2} - 5)}{4} + 2 e^{x} x$

Step 4.6

To write $2 e^{x} x$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{e^{x} (4 x^{2} - 5)}{4} + 2 e^{x} x \cdot \frac{4}{4}$

Step 4.7

Combine $2 e^{x} x$ and $\frac{4}{4}$ .

$\frac{e^{x} (4 x^{2} - 5)}{4} + \frac{2 e^{x} x \cdot 4}{4}$

Step 4.8

Combine the numerators over the common denominator.

$\frac{e^{x} (4 x^{2} - 5) + 2 e^{x} x \cdot 4}{4}$

Step 4.9

Simplify the numerator.

Tap for more steps...

Step 4.9.1

Factor $e^{x}$ out of $e^{x} (4 x^{2} - 5) + 2 e^{x} x \cdot 4$ .

Tap for more steps...

Step 4.9.1.1

Factor $e^{x}$ out of $2 e^{x} x \cdot 4$ .

$\frac{e^{x} (4 x^{2} - 5) + e^{x} (2 x \cdot 4)}{4}$

Step 4.9.1.2

Factor $e^{x}$ out of $e^{x} (4 x^{2} - 5) + e^{x} (2 x \cdot 4)$ .

$\frac{e^{x} (4 x^{2} - 5 + 2 x \cdot 4)}{4}$

Step 4.9.2

Multiply $4$ by $2$ .

$\frac{e^{x} (4 x^{2} - 5 + 8 x)}{4}$

Step 4.9.3

Reorder terms.

$\frac{e^{x} (4 x^{2} + 8 x - 5)}{4}$

Step 4.9.4

Factor by grouping.

Tap for more steps...

Step 4.9.4.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 4 \cdot - 5 = - 20$ and whose sum is $b = 8$ .

Tap for more steps...

Step 4.9.4.1.1

Factor $8$ out of $8 x$ .

$\frac{e^{x} (4 x^{2} + 8 (x) - 5)}{4}$

Step 4.9.4.1.2

Rewrite $8$ as $- 2$ plus $10$

$\frac{e^{x} (4 x^{2} + (- 2 + 10) x - 5)}{4}$

Step 4.9.4.1.3

Apply the distributive property.

$\frac{e^{x} (4 x^{2} - 2 x + 10 x - 5)}{4}$

Step 4.9.4.2

Factor out the greatest common factor from each group.

Tap for more steps...

Step 4.9.4.2.1

Group the first two terms and the last two terms.

$\frac{e^{x} ((4 x^{2} - 2 x) + 10 x - 5)}{4}$

Step 4.9.4.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{e^{x} (2 x (2 x - 1) + 5 (2 x - 1))}{4}$

Step 4.9.4.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$\frac{e^{x} ((2 x - 1) (2 x + 5))}{4}$

$\frac{e^{x} (2 x - 1) (2 x + 5)}{4}$