Calculus Examples

$63 x^{3} \cdot (6 x^{2}) - 24 (2 x)$

Step 1

By the Sum Rule, the derivative of $63 x^{3} \cdot (6 x^{2}) - 24 (2 x)$ with respect to $x$ is $\frac{d}{d x} [63 x^{3} \cdot (6 x^{2})] + \frac{d}{d x} [- 24 (2 x)]$ .

$\frac{d}{d x} [63 x^{3} \cdot (6 x^{2})] + \frac{d}{d x} [- 24 (2 x)]$

Step 2

Evaluate $\frac{d}{d x} [63 x^{3} \cdot (6 x^{2})]$ .

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Step 2.1

Multiply $6$ by $63$ .

$\frac{d}{d x} [378 x^{3} \cdot x^{2}] + \frac{d}{d x} [- 24 (2 x)]$

Step 2.2

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 2.2.1

Move $x^{2}$ .

$\frac{d}{d x} [378 (x^{2} x^{3})] + \frac{d}{d x} [- 24 (2 x)]$

Step 2.2.2

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

$\frac{d}{d x} [378 x^{2 + 3}] + \frac{d}{d x} [- 24 (2 x)]$

Step 2.2.3

Add $2$ and $3$ .

$\frac{d}{d x} [378 x^{5}] + \frac{d}{d x} [- 24 (2 x)]$

Step 2.3

Since $378$ is constant with respect to $x$ , the derivative of $378 x^{5}$ with respect to $x$ is $378 \frac{d}{d x} [x^{5}]$ .

$378 \frac{d}{d x} [x^{5}] + \frac{d}{d x} [- 24 (2 x)]$

Step 2.4

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

$378 (5 x^{4}) + \frac{d}{d x} [- 24 (2 x)]$

Step 2.5

Multiply $5$ by $378$ .

$1890 x^{4} + \frac{d}{d x} [- 24 (2 x)]$

Step 3

Evaluate $\frac{d}{d x} [- 24 (2 x)]$ .

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Step 3.1

Multiply $2$ by $- 24$ .

$1890 x^{4} + \frac{d}{d x} [- 48 x]$

Step 3.2

Since $- 48$ is constant with respect to $x$ , the derivative of $- 48 x$ with respect to $x$ is $- 48 \frac{d}{d x} [x]$ .

$1890 x^{4} - 48 \frac{d}{d x} [x]$

Step 3.3

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

$1890 x^{4} - 48 \cdot 1$

Step 3.4

Multiply $- 48$ by $1$ .

$1890 x^{4} - 48$