Calculus Examples

$y = x^{5} e^{x}$

Step 1

Find the first derivative.

Tap for more steps...

Step 1.1

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 $f (x) = x^{5}$ and $g (x) = e^{x}$ .

$x^{5} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{5}]$

Step 1.2

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

$x^{5} e^{x} + e^{x} \frac{d}{d x} [x^{5}]$

Step 1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$x^{5} e^{x} + e^{x} (5 x^{4})$

Step 1.4

Simplify.

Tap for more steps...

Step 1.4.1

Reorder terms.

$e^{x} x^{5} + 5 e^{x} x^{4}$

Step 1.4.2

Reorder factors in $e^{x} x^{5} + 5 e^{x} x^{4}$ .

$f' (x) = x^{5} e^{x} + 5 x^{4} e^{x}$

Step 2

Find the second derivative.

Tap for more steps...

Step 2.1

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

$\frac{d}{d x} [x^{5} e^{x}] + \frac{d}{d x} [5 x^{4} e^{x}]$

Step 2.2

Evaluate $\frac{d}{d x} [x^{5} e^{x}]$ .

Tap for more steps...

Step 2.2.1

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 $f (x) = x^{5}$ and $g (x) = e^{x}$ .

$x^{5} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [5 x^{4} e^{x}]$

Step 2.2.2

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

$x^{5} e^{x} + e^{x} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [5 x^{4} e^{x}]$

Step 2.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

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

Step 2.3

Evaluate $\frac{d}{d x} [5 x^{4} e^{x}]$ .

Tap for more steps...

Step 2.3.1

Since $5$ is constant with respect to $x$ , the derivative of $5 x^{4} e^{x}$ with respect to $x$ is $5 \frac{d}{d x} [x^{4} e^{x}]$ .

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

Step 2.3.2

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 $f (x) = x^{4}$ and $g (x) = e^{x}$ .

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

Step 2.3.3

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

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

Step 2.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$x^{5} e^{x} + e^{x} (5 x^{4}) + 5 (x^{4} e^{x} + e^{x} (4 x^{3}))$

Step 2.4

Simplify.

Tap for more steps...

Step 2.4.1

Apply the distributive property.

$x^{5} e^{x} + e^{x} (5 x^{4}) + 5 (x^{4} e^{x}) + 5 (e^{x} (4 x^{3}))$

Step 2.4.2

Combine terms.

Tap for more steps...

Step 2.4.2.1

Multiply $4$ by $5$ .

$x^{5} e^{x} + e^{x} (5 x^{4}) + 5 x^{4} e^{x} + 20 (e^{x} (x^{3}))$

Step 2.4.2.2

Add $e^{x} (5 x^{4})$ and $5 x^{4} e^{x}$ .

Tap for more steps...

Step 2.4.2.2.1

Move $e^{x}$ .

$x^{5} e^{x} + 5 x^{4} e^{x} + 5 x^{4} e^{x} + 20 e^{x} x^{3}$

Step 2.4.2.2.2

Add $5 x^{4} e^{x}$ and $5 x^{4} e^{x}$ .

$x^{5} e^{x} + 10 x^{4} e^{x} + 20 e^{x} x^{3}$

Step 2.4.3

Reorder terms.

$e^{x} x^{5} + 10 e^{x} x^{4} + 20 e^{x} x^{3}$

Step 2.4.4

Reorder factors in $e^{x} x^{5} + 10 e^{x} x^{4} + 20 e^{x} x^{3}$ .

$f'' (x) = x^{5} e^{x} + 10 x^{4} e^{x} + 20 x^{3} e^{x}$

Step 3

Find the third derivative.

Tap for more steps...

Step 3.1

By the Sum Rule, the derivative of $x^{5} e^{x} + 10 x^{4} e^{x} + 20 x^{3} e^{x}$ with respect to $x$ is $\frac{d}{d x} [x^{5} e^{x}] + \frac{d}{d x} [10 x^{4} e^{x}] + \frac{d}{d x} [20 x^{3} e^{x}]$ .

$\frac{d}{d x} [x^{5} e^{x}] + \frac{d}{d x} [10 x^{4} e^{x}] + \frac{d}{d x} [20 x^{3} e^{x}]$

Step 3.2

Evaluate $\frac{d}{d x} [x^{5} e^{x}]$ .

Tap for more steps...

Step 3.2.1

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 $f (x) = x^{5}$ and $g (x) = e^{x}$ .

$x^{5} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [10 x^{4} e^{x}] + \frac{d}{d x} [20 x^{3} e^{x}]$

Step 3.2.2

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

$x^{5} e^{x} + e^{x} \frac{d}{d x} [x^{5}] + \frac{d}{d x} [10 x^{4} e^{x}] + \frac{d}{d x} [20 x^{3} e^{x}]$

Step 3.2.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 5$ .

$x^{5} e^{x} + e^{x} (5 x^{4}) + \frac{d}{d x} [10 x^{4} e^{x}] + \frac{d}{d x} [20 x^{3} e^{x}]$

Step 3.3

Evaluate $\frac{d}{d x} [10 x^{4} e^{x}]$ .

Tap for more steps...

Step 3.3.1

Since $10$ is constant with respect to $x$ , the derivative of $10 x^{4} e^{x}$ with respect to $x$ is $10 \frac{d}{d x} [x^{4} e^{x}]$ .

$x^{5} e^{x} + e^{x} (5 x^{4}) + 10 \frac{d}{d x} [x^{4} e^{x}] + \frac{d}{d x} [20 x^{3} e^{x}]$

Step 3.3.2

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 $f (x) = x^{4}$ and $g (x) = e^{x}$ .

$x^{5} e^{x} + e^{x} (5 x^{4}) + 10 (x^{4} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [20 x^{3} e^{x}]$

Step 3.3.3

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

$x^{5} e^{x} + e^{x} (5 x^{4}) + 10 (x^{4} e^{x} + e^{x} \frac{d}{d x} [x^{4}]) + \frac{d}{d x} [20 x^{3} e^{x}]$

Step 3.3.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$x^{5} e^{x} + e^{x} (5 x^{4}) + 10 (x^{4} e^{x} + e^{x} (4 x^{3})) + \frac{d}{d x} [20 x^{3} e^{x}]$

Step 3.4

Evaluate $\frac{d}{d x} [20 x^{3} e^{x}]$ .

Tap for more steps...

Step 3.4.1

Since $20$ is constant with respect to $x$ , the derivative of $20 x^{3} e^{x}$ with respect to $x$ is $20 \frac{d}{d x} [x^{3} e^{x}]$ .

$x^{5} e^{x} + e^{x} (5 x^{4}) + 10 (x^{4} e^{x} + e^{x} (4 x^{3})) + 20 \frac{d}{d x} [x^{3} e^{x}]$

Step 3.4.2

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 $f (x) = x^{3}$ and $g (x) = e^{x}$ .

$x^{5} e^{x} + e^{x} (5 x^{4}) + 10 (x^{4} e^{x} + e^{x} (4 x^{3})) + 20 (x^{3} \frac{d}{d x} [e^{x}] + e^{x} \frac{d}{d x} [x^{3}])$

Step 3.4.3

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

$x^{5} e^{x} + e^{x} (5 x^{4}) + 10 (x^{4} e^{x} + e^{x} (4 x^{3})) + 20 (x^{3} e^{x} + e^{x} \frac{d}{d x} [x^{3}])$

Step 3.4.4

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$x^{5} e^{x} + e^{x} (5 x^{4}) + 10 (x^{4} e^{x} + e^{x} (4 x^{3})) + 20 (x^{3} e^{x} + e^{x} (3 x^{2}))$

Step 3.5

Simplify.

Tap for more steps...

Step 3.5.1

Apply the distributive property.

$x^{5} e^{x} + e^{x} (5 x^{4}) + 10 (x^{4} e^{x}) + 10 (e^{x} (4 x^{3})) + 20 (x^{3} e^{x} + e^{x} (3 x^{2}))$

Step 3.5.2

Apply the distributive property.

$x^{5} e^{x} + e^{x} (5 x^{4}) + 10 (x^{4} e^{x}) + 10 (e^{x} (4 x^{3})) + 20 (x^{3} e^{x}) + 20 (e^{x} (3 x^{2}))$

Step 3.5.3

Combine terms.

Tap for more steps...

Step 3.5.3.1

Multiply $4$ by $10$ .

$x^{5} e^{x} + e^{x} (5 x^{4}) + 10 x^{4} e^{x} + 40 (e^{x} (x^{3})) + 20 (x^{3} e^{x}) + 20 (e^{x} (3 x^{2}))$

Step 3.5.3.2

Add $e^{x} (5 x^{4})$ and $10 x^{4} e^{x}$ .

Tap for more steps...

Step 3.5.3.2.1

Move $e^{x}$ .

$x^{5} e^{x} + 5 x^{4} e^{x} + 10 x^{4} e^{x} + 40 e^{x} x^{3} + 20 (x^{3} e^{x}) + 20 (e^{x} (3 x^{2}))$

Step 3.5.3.2.2

Add $5 x^{4} e^{x}$ and $10 x^{4} e^{x}$ .

$x^{5} e^{x} + 15 x^{4} e^{x} + 40 e^{x} x^{3} + 20 (x^{3} e^{x}) + 20 (e^{x} (3 x^{2}))$

Step 3.5.3.3

Multiply $3$ by $20$ .

$x^{5} e^{x} + 15 x^{4} e^{x} + 40 e^{x} x^{3} + 20 x^{3} e^{x} + 60 (e^{x} (x^{2}))$

Step 3.5.3.4

Add $40 e^{x} x^{3}$ and $20 x^{3} e^{x}$ .

Tap for more steps...

Step 3.5.3.4.1

Move $e^{x}$ .

$x^{5} e^{x} + 15 x^{4} e^{x} + 40 x^{3} e^{x} + 20 x^{3} e^{x} + 60 e^{x} x^{2}$

Step 3.5.3.4.2

Add $40 x^{3} e^{x}$ and $20 x^{3} e^{x}$ .

$x^{5} e^{x} + 15 x^{4} e^{x} + 60 x^{3} e^{x} + 60 e^{x} x^{2}$

Step 3.5.4

Reorder terms.

$e^{x} x^{5} + 15 e^{x} x^{4} + 60 e^{x} x^{3} + 60 e^{x} x^{2}$

Step 3.5.5

Reorder factors in $e^{x} x^{5} + 15 e^{x} x^{4} + 60 e^{x} x^{3} + 60 e^{x} x^{2}$ .

$f''' (x) = x^{5} e^{x} + 15 x^{4} e^{x} + 60 x^{3} e^{x} + 60 x^{2} e^{x}$