Calculus Examples

$y = x^{9} \cdot x^{4}$

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 $f (x) = x^{9}$ and $g (x) = x^{4}$ .

$x^{9} \frac{d}{d x} [x^{4}] + x^{4} \frac{d}{d x} [x^{9}]$

Step 2

Differentiate using the Power Rule.

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Step 2.1

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

$x^{9} (4 x^{3}) + x^{4} \frac{d}{d x} [x^{9}]$

Step 2.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 9$ .

$x^{9} (4 x^{3}) + x^{4} (9 x^{8})$

Step 3

Combine terms.

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Step 3.1

Multiply $x^{9}$ by $x^{3}$ by adding the exponents.

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Step 3.1.1

Move $x^{3}$ .

$x^{3} x^{9} \cdot 4 + x^{4} (9 x^{8})$

Step 3.1.2

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

$x^{3 + 9} \cdot 4 + x^{4} (9 x^{8})$

Step 3.1.3

Add $3$ and $9$ .

$x^{12} \cdot 4 + x^{4} (9 x^{8})$

Step 3.2

Move $4$ to the left of $x^{12}$ .

$4 \cdot x^{12} + x^{4} (9 x^{8})$

Step 3.3

Multiply $x^{4}$ by $x^{8}$ by adding the exponents.

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Step 3.3.1

Move $x^{8}$ .

$4 x^{12} + x^{8} x^{4} \cdot 9$

Step 3.3.2

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

$4 x^{12} + x^{8 + 4} \cdot 9$

Step 3.3.3

Add $8$ and $4$ .

$4 x^{12} + x^{12} \cdot 9$

Step 3.4

Move $9$ to the left of $x^{12}$ .

$4 x^{12} + 9 \cdot x^{12}$

Step 3.5

Add $4 x^{12}$ and $9 x^{12}$ .

$13 x^{12}$