Calculus Examples

${({(x^{2} + 3)}^{5} + x)}^{2}$

Step 1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{2}$ and $g (x) = {(x^{2} + 3)}^{5} + x$ .

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Step 1.1

To apply the Chain Rule, set $u_{1}$ as ${(x^{2} + 3)}^{5} + x$ .

$\frac{d}{d u_{1}} [{u_{1}}^{2}] \frac{d}{d x} [{(x^{2} + 3)}^{5} + x]$

Step 1.2

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

$2 u_{1} \frac{d}{d x} [{(x^{2} + 3)}^{5} + x]$

Step 1.3

Replace all occurrences of $u_{1}$ with ${(x^{2} + 3)}^{5} + x$ .

$2 ({(x^{2} + 3)}^{5} + x) \frac{d}{d x} [{(x^{2} + 3)}^{5} + x]$

Step 2

By the Sum Rule, the derivative of ${(x^{2} + 3)}^{5} + x$ with respect to $x$ is $\frac{d}{d x} [{(x^{2} + 3)}^{5}] + \frac{d}{d x} [x]$ .

$2 ({(x^{2} + 3)}^{5} + x) (\frac{d}{d x} [{(x^{2} + 3)}^{5}] + \frac{d}{d x} [x])$

Step 3

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 $g (x) = x^{2} + 3$ .

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Step 3.1

To apply the Chain Rule, set $u_{2}$ as $x^{2} + 3$ .

$2 ({(x^{2} + 3)}^{5} + x) (\frac{d}{d u_{2}} [{u_{2}}^{5}] \frac{d}{d x} [x^{2} + 3] + \frac{d}{d x} [x])$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u_{2}} [{u_{2}}^{n}]$ is $n {u_{2}}^{n - 1}$ where $n = 5$ .

$2 ({(x^{2} + 3)}^{5} + x) (5 {u_{2}}^{4} \frac{d}{d x} [x^{2} + 3] + \frac{d}{d x} [x])$

Step 3.3

Replace all occurrences of $u_{2}$ with $x^{2} + 3$ .

$2 ({(x^{2} + 3)}^{5} + x) (5 {(x^{2} + 3)}^{4} \frac{d}{d x} [x^{2} + 3] + \frac{d}{d x} [x])$

Step 4

Differentiate.

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Step 4.1

By the Sum Rule, the derivative of $x^{2} + 3$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]$ .

$2 ({(x^{2} + 3)}^{5} + x) (5 {(x^{2} + 3)}^{4} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [3]) + \frac{d}{d x} [x])$

Step 4.2

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

$2 ({(x^{2} + 3)}^{5} + x) (5 {(x^{2} + 3)}^{4} (2 x + \frac{d}{d x} [3]) + \frac{d}{d x} [x])$

Step 4.3

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

$2 ({(x^{2} + 3)}^{5} + x) (5 {(x^{2} + 3)}^{4} (2 x + 0) + \frac{d}{d x} [x])$

Step 4.4

Simplify the expression.

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Step 4.4.1

Add $2 x$ and $0$ .

$2 ({(x^{2} + 3)}^{5} + x) (5 {(x^{2} + 3)}^{4} (2 x) + \frac{d}{d x} [x])$

Step 4.4.2

Multiply $2$ by $5$ .

$2 ({(x^{2} + 3)}^{5} + x) (10 {(x^{2} + 3)}^{4} x + \frac{d}{d x} [x])$

Step 4.5

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

$2 ({(x^{2} + 3)}^{5} + x) (10 {(x^{2} + 3)}^{4} x + 1)$

Step 5

Simplify.

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Step 5.1

Apply the distributive property.

$(2 {(x^{2} + 3)}^{5} + 2 x) (10 {(x^{2} + 3)}^{4} x + 1)$

Step 5.2

Reorder the factors of $(2 {(x^{2} + 3)}^{5} + 2 x) (10 {(x^{2} + 3)}^{4} x + 1)$ .

$(10 {(x^{2} + 3)}^{4} x + 1) (2 {(x^{2} + 3)}^{5} + 2 x)$