Algebra Examples

$10 e^{10 x} x^{2} + 2 e^{10 x} x$

Step 1

By the Sum Rule, the derivative of $10 e^{10 x} x^{2} + 2 e^{10 x} x$ with respect to $x$ is $\frac{d}{d x} [10 e^{10 x} x^{2}] + \frac{d}{d x} [2 e^{10 x} x]$ .

$\frac{d}{d x} [10 e^{10 x} x^{2}] + \frac{d}{d x} [2 e^{10 x} x]$

Step 2

Evaluate $\frac{d}{d x} [10 e^{10 x} x^{2}]$ .

Tap for more steps...

Step 2.1

Since $10$ is constant with respect to $x$ , the derivative of $10 e^{10 x} x^{2}$ with respect to $x$ is $10 \frac{d}{d x} [e^{10 x} x^{2}]$ .

$10 \frac{d}{d x} [e^{10 x} x^{2}] + \frac{d}{d x} [2 e^{10 x} x]$

Step 2.2

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 $f (x) = e^{10 x}$ and $g (x) = x^{2}$ .

$10 (e^{10 x} \frac{d}{d x} [x^{2}] + x^{2} \frac{d}{d x} [e^{10 x}]) + \frac{d}{d x} [2 e^{10 x} x]$

Step 2.3

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

$10 (e^{10 x} (2 x) + x^{2} \frac{d}{d x} [e^{10 x}]) + \frac{d}{d x} [2 e^{10 x} x]$

Step 2.4

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 $g (x) = 10 x$ .

Tap for more steps...

Step 2.4.1

To apply the Chain Rule, set $u_{1}$ as $10 x$ .

$10 (e^{10 x} (2 x) + x^{2} (\frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [10 x])) + \frac{d}{d x} [2 e^{10 x} x]$

Step 2.4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$10 (e^{10 x} (2 x) + x^{2} (e^{u_{1}} \frac{d}{d x} [10 x])) + \frac{d}{d x} [2 e^{10 x} x]$

Step 2.4.3

Replace all occurrences of $u_{1}$ with $10 x$ .

$10 (e^{10 x} (2 x) + x^{2} (e^{10 x} \frac{d}{d x} [10 x])) + \frac{d}{d x} [2 e^{10 x} x]$

Step 2.5

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

$10 (e^{10 x} (2 x) + x^{2} (e^{10 x} (10 \frac{d}{d x} [x]))) + \frac{d}{d x} [2 e^{10 x} x]$

Step 2.6

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

$10 (e^{10 x} (2 x) + x^{2} (e^{10 x} (10 \cdot 1))) + \frac{d}{d x} [2 e^{10 x} x]$

Step 2.7

Multiply $10$ by $1$ .

$10 (e^{10 x} (2 x) + x^{2} (e^{10 x} \cdot 10)) + \frac{d}{d x} [2 e^{10 x} x]$

Step 2.8

Move $10$ to the left of $e^{10 x}$ .

$10 (e^{10 x} (2 x) + x^{2} (10 e^{10 x})) + \frac{d}{d x} [2 e^{10 x} x]$

Step 3

Evaluate $\frac{d}{d x} [2 e^{10 x} x]$ .

Tap for more steps...

Step 3.1

Since $2$ is constant with respect to $x$ , the derivative of $2 e^{10 x} x$ with respect to $x$ is $2 \frac{d}{d x} [e^{10 x} x]$ .

$10 (e^{10 x} (2 x) + x^{2} (10 e^{10 x})) + 2 \frac{d}{d x} [e^{10 x} x]$

Step 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 $f (x) = e^{10 x}$ and $g (x) = x$ .

$10 (e^{10 x} (2 x) + x^{2} (10 e^{10 x})) + 2 (e^{10 x} \frac{d}{d x} [x] + x \frac{d}{d x} [e^{10 x}])$

Step 3.3

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

$10 (e^{10 x} (2 x) + x^{2} (10 e^{10 x})) + 2 (e^{10 x} \cdot 1 + x \frac{d}{d x} [e^{10 x}])$

Step 3.4

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 $g (x) = 10 x$ .

Tap for more steps...

Step 3.4.1

To apply the Chain Rule, set $u_{2}$ as $10 x$ .

$10 (e^{10 x} (2 x) + x^{2} (10 e^{10 x})) + 2 (e^{10 x} \cdot 1 + x (\frac{d}{d u_{2}} [e^{u_{2}}] \frac{d}{d x} [10 x]))$

Step 3.4.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{2}} [a^{u_{2}}]$ is $a^{u_{2}} \ln (a)$ where $a$ = $e$ .

$10 (e^{10 x} (2 x) + x^{2} (10 e^{10 x})) + 2 (e^{10 x} \cdot 1 + x (e^{u_{2}} \frac{d}{d x} [10 x]))$

Step 3.4.3

Replace all occurrences of $u_{2}$ with $10 x$ .

$10 (e^{10 x} (2 x) + x^{2} (10 e^{10 x})) + 2 (e^{10 x} \cdot 1 + x (e^{10 x} \frac{d}{d x} [10 x]))$

Step 3.5

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

$10 (e^{10 x} (2 x) + x^{2} (10 e^{10 x})) + 2 (e^{10 x} \cdot 1 + x (e^{10 x} (10 \frac{d}{d x} [x])))$

Step 3.6

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

$10 (e^{10 x} (2 x) + x^{2} (10 e^{10 x})) + 2 (e^{10 x} \cdot 1 + x (e^{10 x} (10 \cdot 1)))$

Step 3.7

Multiply $e^{10 x}$ by $1$ .

$10 (e^{10 x} (2 x) + x^{2} (10 e^{10 x})) + 2 (e^{10 x} + x (e^{10 x} (10 \cdot 1)))$

Step 3.8

Multiply $10$ by $1$ .

$10 (e^{10 x} (2 x) + x^{2} (10 e^{10 x})) + 2 (e^{10 x} + x (e^{10 x} \cdot 10))$

Step 3.9

Move $10$ to the left of $e^{10 x}$ .

$10 (e^{10 x} (2 x) + x^{2} (10 e^{10 x})) + 2 (e^{10 x} + x (10 e^{10 x}))$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

$10 (e^{10 x} (2 x)) + 10 (x^{2} (10 e^{10 x})) + 2 (e^{10 x} + x (10 e^{10 x}))$

Step 4.2

Apply the distributive property.

$10 (e^{10 x} (2 x)) + 10 (x^{2} (10 e^{10 x})) + 2 e^{10 x} + 2 (x (10 e^{10 x}))$

Step 4.3

Combine terms.

Tap for more steps...

Step 4.3.1

Multiply $2$ by $10$ .

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

Step 4.3.2

Multiply $10$ by $10$ .

$20 e^{10 x} x + 100 (x^{2} (e^{10 x})) + 2 e^{10 x} + 2 (x (10 e^{10 x}))$

Step 4.3.3

Multiply $10$ by $2$ .

$20 e^{10 x} x + 100 x^{2} e^{10 x} + 2 e^{10 x} + 20 (x (e^{10 x}))$

Step 4.3.4

Add $20 e^{10 x} x$ and $20 x e^{10 x}$ .

Tap for more steps...

Step 4.3.4.1

Move $e^{10 x}$ .

$20 x e^{10 x} + 20 x e^{10 x} + 100 x^{2} e^{10 x} + 2 e^{10 x}$

Step 4.3.4.2

Add $20 x e^{10 x}$ and $20 x e^{10 x}$ .

$40 x e^{10 x} + 100 x^{2} e^{10 x} + 2 e^{10 x}$

Step 4.4

Reorder terms.

$40 e^{10 x} x + 100 e^{10 x} x^{2} + 2 e^{10 x}$

Step 4.5

Reorder factors in $40 e^{10 x} x + 100 e^{10 x} x^{2} + 2 e^{10 x}$ .

$40 x e^{10 x} + 100 x^{2} e^{10 x} + 2 e^{10 x}$