Calculus Examples

$y = 3 x (6 x - 5 x^{2})$

Step 1

Since $3$ is constant with respect to $x$ , the derivative of $3 x (6 x - 5 x^{2})$ with respect to $x$ is $3 \frac{d}{d x} [x (6 x - 5 x^{2})]$ .

$3 \frac{d}{d x} [x (6 x - 5 x^{2})]$

Step 2

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 $g (x) = 6 x - 5 x^{2}$ .

$3 (x \frac{d}{d x} [6 x - 5 x^{2}] + (6 x - 5 x^{2}) \frac{d}{d x} [x])$

Step 3

Differentiate.

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Step 3.1

By the Sum Rule, the derivative of $6 x - 5 x^{2}$ with respect to $x$ is $\frac{d}{d x} [6 x] + \frac{d}{d x} [- 5 x^{2}]$ .

$3 (x (\frac{d}{d x} [6 x] + \frac{d}{d x} [- 5 x^{2}]) + (6 x - 5 x^{2}) \frac{d}{d x} [x])$

Step 3.2

Since $6$ is constant with respect to $x$ , the derivative of $6 x$ with respect to $x$ is $6 \frac{d}{d x} [x]$ .

$3 (x (6 \frac{d}{d x} [x] + \frac{d}{d x} [- 5 x^{2}]) + (6 x - 5 x^{2}) \frac{d}{d x} [x])$

Step 3.3

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

$3 (x (6 \cdot 1 + \frac{d}{d x} [- 5 x^{2}]) + (6 x - 5 x^{2}) \frac{d}{d x} [x])$

Step 3.4

Multiply $6$ by $1$ .

$3 (x (6 + \frac{d}{d x} [- 5 x^{2}]) + (6 x - 5 x^{2}) \frac{d}{d x} [x])$

Step 3.5

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5 x^{2}$ with respect to $x$ is $- 5 \frac{d}{d x} [x^{2}]$ .

$3 (x (6 - 5 \frac{d}{d x} [x^{2}]) + (6 x - 5 x^{2}) \frac{d}{d x} [x])$

Step 3.6

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

$3 (x (6 - 5 (2 x)) + (6 x - 5 x^{2}) \frac{d}{d x} [x])$

Step 3.7

Multiply $2$ by $- 5$ .

$3 (x (6 - 10 x) + (6 x - 5 x^{2}) \frac{d}{d x} [x])$

Step 3.8

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

$3 (x (6 - 10 x) + (6 x - 5 x^{2}) \cdot 1)$

Step 3.9

Multiply $6 x - 5 x^{2}$ by $1$ .

$3 (x (6 - 10 x) + 6 x - 5 x^{2})$

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

$3 (x \cdot 6 + x (- 10 x) + 6 x - 5 x^{2})$

Step 4.2

Apply the distributive property.

$3 (x \cdot 6) + 3 (x (- 10 x)) + 3 (6 x) + 3 (- 5 x^{2})$

Step 4.3

Combine terms.

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Step 4.3.1

Move $6$ to the left of $x$ .

$3 (6 \cdot x) + 3 (x (- 10 x)) + 3 (6 x) + 3 (- 5 x^{2})$

Step 4.3.2

Multiply $6$ by $3$ .

$18 x + 3 (x (- 10 x)) + 3 (6 x) + 3 (- 5 x^{2})$

Step 4.3.3

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

$18 x + 3 (- 10 (x^{1} x)) + 3 (6 x) + 3 (- 5 x^{2})$

Step 4.3.4

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

$18 x + 3 (- 10 (x^{1} x^{1})) + 3 (6 x) + 3 (- 5 x^{2})$

Step 4.3.5

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

$18 x + 3 (- 10 x^{1 + 1}) + 3 (6 x) + 3 (- 5 x^{2})$

Step 4.3.6

Add $1$ and $1$ .

$18 x + 3 (- 10 x^{2}) + 3 (6 x) + 3 (- 5 x^{2})$

Step 4.3.7

Multiply $- 10$ by $3$ .

$18 x - 30 x^{2} + 3 (6 x) + 3 (- 5 x^{2})$

Step 4.3.8

Multiply $6$ by $3$ .

$18 x - 30 x^{2} + 18 x + 3 (- 5 x^{2})$

Step 4.3.9

Add $18 x$ and $18 x$ .

$- 30 x^{2} + 36 x + 3 (- 5 x^{2})$

Step 4.3.10

Multiply $- 5$ by $3$ .

$- 30 x^{2} + 36 x - 15 x^{2}$

Step 4.3.11

Subtract $15 x^{2}$ from $- 30 x^{2}$ .

$- 45 x^{2} + 36 x$