Calculus Examples

$y = x {(x^{2} + 1)}^{5}$

Step 1

Differentiate both sides of the equation.

$\frac{d}{d x} (y) = \frac{d}{d x} (x {(x^{2} + 1)}^{5})$

Step 2

The derivative of $y$ with respect to $x$ is $y'$ .

$y'$

Step 3

Differentiate the right side of the equation.

Tap for more steps...

Step 3.1

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 $f (x) = x$ and $g (x) = {(x^{2} + 1)}^{5}$ .

$x \frac{d}{d x} [{(x^{2} + 1)}^{5}] + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.2

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{5}$ and $g (x) = x^{2} + 1$ .

Tap for more steps...

Step 3.2.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

$x (\frac{d}{d u} [u^{5}] \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.2.2

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

$x (5 u^{4} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.2.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

$x (5 {(x^{2} + 1)}^{4} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.3

Differentiate.

Tap for more steps...

Step 3.3.1

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$x (5 {(x^{2} + 1)}^{4} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1])) + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.3.2

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

$x (5 {(x^{2} + 1)}^{4} (2 x + \frac{d}{d x} [1])) + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.3.3

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$x (5 {(x^{2} + 1)}^{4} (2 x + 0)) + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.3.4

Simplify the expression.

Tap for more steps...

Step 3.3.4.1

Add $2 x$ and $0$ .

$x (5 {(x^{2} + 1)}^{4} (2 x)) + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.3.4.2

Multiply $2$ by $5$ .

$x (10 {(x^{2} + 1)}^{4} x) + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.4

Raise $x$ to the power of $1$ .

$x^{1} x (10 {(x^{2} + 1)}^{4}) + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.5

Raise $x$ to the power of $1$ .

$x^{1} x^{1} (10 {(x^{2} + 1)}^{4}) + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.6

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

$x^{1 + 1} (10 {(x^{2} + 1)}^{4}) + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.7

Add $1$ and $1$ .

$x^{2} (10 {(x^{2} + 1)}^{4}) + {(x^{2} + 1)}^{5} \frac{d}{d x} [x]$

Step 3.8

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

$x^{2} (10 {(x^{2} + 1)}^{4}) + {(x^{2} + 1)}^{5} \cdot 1$

Step 3.9

Multiply ${(x^{2} + 1)}^{5}$ by $1$ .

$x^{2} (10 {(x^{2} + 1)}^{4}) + {(x^{2} + 1)}^{5}$

Step 3.10

Simplify.

Tap for more steps...

Step 3.10.1

Factor ${(x^{2} + 1)}^{4}$ out of $x^{2} (10 {(x^{2} + 1)}^{4}) + {(x^{2} + 1)}^{5}$ .

Tap for more steps...

Step 3.10.1.1

Factor ${(x^{2} + 1)}^{4}$ out of $x^{2} (10 {(x^{2} + 1)}^{4})$ .

${(x^{2} + 1)}^{4} (x^{2} \cdot (10)) + {(x^{2} + 1)}^{5}$

Step 3.10.1.2

Factor ${(x^{2} + 1)}^{4}$ out of ${(x^{2} + 1)}^{5}$ .

${(x^{2} + 1)}^{4} (x^{2} \cdot (10)) + {(x^{2} + 1)}^{4} (x^{2} + 1)$

Step 3.10.1.3

Factor ${(x^{2} + 1)}^{4}$ out of ${(x^{2} + 1)}^{4} (x^{2} \cdot (10)) + {(x^{2} + 1)}^{4} (x^{2} + 1)$ .

${(x^{2} + 1)}^{4} (x^{2} \cdot (10) + x^{2} + 1)$

Step 3.10.2

Combine terms.

Tap for more steps...

Step 3.10.2.1

Move $10$ to the left of $x^{2}$ .

${(x^{2} + 1)}^{4} (10 \cdot x^{2} + x^{2} + 1)$

Step 3.10.2.2

Add $10 x^{2}$ and $x^{2}$ .

${(x^{2} + 1)}^{4} (11 x^{2} + 1)$

Step 4

Reform the equation by setting the left side equal to the right side.

$y' = {(x^{2} + 1)}^{4} (11 x^{2} + 1)$

Step 5

Replace $y'$ with $\frac{d y}{d x}$ .

$\frac{d y}{d x} = {(x^{2} + 1)}^{4} (11 x^{2} + 1)$