Calculus Examples

$y = x {(x^{3} + 7)}^{5}$

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 $f (x) = x$ and $g (x) = {(x^{3} + 7)}^{5}$ .

$x \frac{d}{d x} [{(x^{3} + 7)}^{5}] + {(x^{3} + 7)}^{5} \frac{d}{d x} [x]$

Step 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 $g (x) = x^{3} + 7$ .

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Step 2.1

To apply the Chain Rule, set $u$ as $x^{3} + 7$ .

$x (\frac{d}{d u} [u^{5}] \frac{d}{d x} [x^{3} + 7]) + {(x^{3} + 7)}^{5} \frac{d}{d x} [x]$

Step 2.2

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

$x (5 u^{4} \frac{d}{d x} [x^{3} + 7]) + {(x^{3} + 7)}^{5} \frac{d}{d x} [x]$

Step 2.3

Replace all occurrences of $u$ with $x^{3} + 7$ .

$x (5 {(x^{3} + 7)}^{4} \frac{d}{d x} [x^{3} + 7]) + {(x^{3} + 7)}^{5} \frac{d}{d x} [x]$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $x^{3} + 7$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [7]$ .

$x (5 {(x^{3} + 7)}^{4} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [7])) + {(x^{3} + 7)}^{5} \frac{d}{d x} [x]$

Step 3.2

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

$x (5 {(x^{3} + 7)}^{4} (3 x^{2} + \frac{d}{d x} [7])) + {(x^{3} + 7)}^{5} \frac{d}{d x} [x]$

Step 3.3

Since $7$ is constant with respect to $x$ , the derivative of $7$ with respect to $x$ is $0$ .

$x (5 {(x^{3} + 7)}^{4} (3 x^{2} + 0)) + {(x^{3} + 7)}^{5} \frac{d}{d x} [x]$

Step 3.4

Simplify the expression.

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Step 3.4.1

Add $3 x^{2}$ and $0$ .

$x (5 {(x^{3} + 7)}^{4} (3 x^{2})) + {(x^{3} + 7)}^{5} \frac{d}{d x} [x]$

Step 3.4.2

Multiply $3$ by $5$ .

$x (15 {(x^{3} + 7)}^{4} x^{2}) + {(x^{3} + 7)}^{5} \frac{d}{d x} [x]$

Step 4

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

$x^{2} x^{1} (15 {(x^{3} + 7)}^{4}) + {(x^{3} + 7)}^{5} \frac{d}{d x} [x]$

Step 5

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

$x^{2 + 1} (15 {(x^{3} + 7)}^{4}) + {(x^{3} + 7)}^{5} \frac{d}{d x} [x]$

Step 6

Add $2$ and $1$ .

$x^{3} (15 {(x^{3} + 7)}^{4}) + {(x^{3} + 7)}^{5} \frac{d}{d x} [x]$

Step 7

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

$x^{3} (15 {(x^{3} + 7)}^{4}) + {(x^{3} + 7)}^{5} \cdot 1$

Step 8

Multiply ${(x^{3} + 7)}^{5}$ by $1$ .

$x^{3} (15 {(x^{3} + 7)}^{4}) + {(x^{3} + 7)}^{5}$

Step 9

Simplify.

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Step 9.1

Factor ${(x^{3} + 7)}^{4}$ out of $x^{3} (15 {(x^{3} + 7)}^{4}) + {(x^{3} + 7)}^{5}$ .

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Step 9.1.1

Factor ${(x^{3} + 7)}^{4}$ out of $x^{3} (15 {(x^{3} + 7)}^{4})$ .

${(x^{3} + 7)}^{4} (x^{3} \cdot (15)) + {(x^{3} + 7)}^{5}$

Step 9.1.2

Factor ${(x^{3} + 7)}^{4}$ out of ${(x^{3} + 7)}^{5}$ .

${(x^{3} + 7)}^{4} (x^{3} \cdot (15)) + {(x^{3} + 7)}^{4} (x^{3} + 7)$

Step 9.1.3

Factor ${(x^{3} + 7)}^{4}$ out of ${(x^{3} + 7)}^{4} (x^{3} \cdot (15)) + {(x^{3} + 7)}^{4} (x^{3} + 7)$ .

${(x^{3} + 7)}^{4} (x^{3} \cdot (15) + x^{3} + 7)$

Step 9.2

Combine terms.

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Step 9.2.1

Move $15$ to the left of $x^{3}$ .

${(x^{3} + 7)}^{4} (15 \cdot x^{3} + x^{3} + 7)$

Step 9.2.2

Add $15 x^{3}$ and $x^{3}$ .

${(x^{3} + 7)}^{4} (16 x^{3} + 7)$