Calculus Examples

$(47 u^{46}) (15 x^{2} + 8 x)$

Step 1

This derivative could not be completed using the chain rule. Mathway will use another method.

Step 2

Since $47 u^{46}$ is constant with respect to $x$ , the derivative of $47 u^{46} (15 x^{2} + 8 x)$ with respect to $x$ is $47 u^{46} \frac{d}{d x} [15 x^{2} + 8 x]$ .

$47 u^{46} \frac{d}{d x} [15 x^{2} + 8 x]$

Step 3

By the Sum Rule, the derivative of $15 x^{2} + 8 x$ with respect to $x$ is $\frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [8 x]$ .

$47 u^{46} (\frac{d}{d x} [15 x^{2}] + \frac{d}{d x} [8 x])$

Step 4

Since $15$ is constant with respect to $x$ , the derivative of $15 x^{2}$ with respect to $x$ is $15 \frac{d}{d x} [x^{2}]$ .

$47 u^{46} (15 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [8 x])$

Step 5

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

$47 u^{46} (15 (2 x) + \frac{d}{d x} [8 x])$

Step 6

Multiply $2$ by $15$ .

$47 u^{46} (30 x + \frac{d}{d x} [8 x])$

Step 7

Since $8$ is constant with respect to $x$ , the derivative of $8 x$ with respect to $x$ is $8 \frac{d}{d x} [x]$ .

$47 u^{46} (30 x + 8 \frac{d}{d x} [x])$

Step 8

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

$47 u^{46} (30 x + 8 \cdot 1)$

Step 9

Multiply $8$ by $1$ .

$47 u^{46} (30 x + 8)$

Step 10

Simplify.

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Step 10.1

Apply the distributive property.

$47 u^{46} (30 x) + 47 u^{46} \cdot 8$

Step 10.2

Combine terms.

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Step 10.2.1

Multiply $30$ by $47$ .

$1410 u^{46} x + 47 u^{46} \cdot 8$

Step 10.2.2

Multiply $8$ by $47$ .

$1410 u^{46} x + 376 u^{46}$

Step 10.3

Reorder terms.

$376 u^{46} + 1410 u^{46} x$