Find the Derivative - d/dx x(15-2x)(27-2x)

$x (15 - 2 x) (27 - 2 x)$

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 $f (x) = x (15 - 2 x)$ and $g (x) = 27 - 2 x$ .

$x (15 - 2 x) \frac{d}{d x} [27 - 2 x] + (27 - 2 x) \frac{d}{d x} [x (15 - 2 x)]$

Step 2

Differentiate.

By the Sum Rule, the derivative of $27 - 2 x$ with respect to $x$ is $\frac{d}{d x} [27] + \frac{d}{d x} [- 2 x]$ .

$x (15 - 2 x) (\frac{d}{d x} [27] + \frac{d}{d x} [- 2 x]) + (27 - 2 x) \frac{d}{d x} [x (15 - 2 x)]$

Since $27$ is constant with respect to $x$ , the derivative of $27$ with respect to $x$ is $0$ .

$x (15 - 2 x) (0 + \frac{d}{d x} [- 2 x]) + (27 - 2 x) \frac{d}{d x} [x (15 - 2 x)]$

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$x (15 - 2 x) \frac{d}{d x} [- 2 x] + (27 - 2 x) \frac{d}{d x} [x (15 - 2 x)]$

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

$x (15 - 2 x) (- 2 \frac{d}{d x} [x]) + (27 - 2 x) \frac{d}{d x} [x (15 - 2 x)]$

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

$x (15 - 2 x) (- 2 \cdot 1) + (27 - 2 x) \frac{d}{d x} [x (15 - 2 x)]$

Simplify the expression.

Multiply $- 2$ by $1$ .

$x (15 - 2 x) \cdot - 2 + (27 - 2 x) \frac{d}{d x} [x (15 - 2 x)]$

Move $- 2$ to the left of $x (15 - 2 x)$ .

$- 2 \cdot (x (15 - 2 x)) + (27 - 2 x) \frac{d}{d x} [x (15 - 2 x)]$

Step 3

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 $g (x) = 15 - 2 x$ .

$- 2 x (15 - 2 x) + (27 - 2 x) (x \frac{d}{d x} [15 - 2 x] + (15 - 2 x) \frac{d}{d x} [x])$

Step 4

Differentiate.

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

$- 2 x (15 - 2 x) + (27 - 2 x) (x (\frac{d}{d x} [15] + \frac{d}{d x} [- 2 x]) + (15 - 2 x) \frac{d}{d x} [x])$

Since $15$ is constant with respect to $x$ , the derivative of $15$ with respect to $x$ is $0$ .

$- 2 x (15 - 2 x) + (27 - 2 x) (x (0 + \frac{d}{d x} [- 2 x]) + (15 - 2 x) \frac{d}{d x} [x])$

Add $0$ and $\frac{d}{d x} [- 2 x]$ .

$- 2 x (15 - 2 x) + (27 - 2 x) (x \frac{d}{d x} [- 2 x] + (15 - 2 x) \frac{d}{d x} [x])$

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

$- 2 x (15 - 2 x) + (27 - 2 x) (x (- 2 \frac{d}{d x} [x]) + (15 - 2 x) \frac{d}{d x} [x])$

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

$- 2 x (15 - 2 x) + (27 - 2 x) (x (- 2 \cdot 1) + (15 - 2 x) \frac{d}{d x} [x])$

Simplify the expression.

Multiply $- 2$ by $1$ .

$- 2 x (15 - 2 x) + (27 - 2 x) (x \cdot - 2 + (15 - 2 x) \frac{d}{d x} [x])$

Move $- 2$ to the left of $x$ .

$- 2 x (15 - 2 x) + (27 - 2 x) (- 2 \cdot x + (15 - 2 x) \frac{d}{d x} [x])$

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

$- 2 x (15 - 2 x) + (27 - 2 x) (- 2 x + (15 - 2 x) \cdot 1)$

Simplify by adding terms.

Multiply $15 - 2 x$ by $1$ .

$- 2 x (15 - 2 x) + (27 - 2 x) (- 2 x + 15 - 2 x)$

Subtract $2 x$ from $- 2 x$ .

$- 2 x (15 - 2 x) + (27 - 2 x) (- 4 x + 15)$

Step 5

Simplify.

Apply the distributive property.

$- 2 x \cdot 15 - 2 x (- 2 x) + (27 - 2 x) (- 4 x + 15)$

Apply the distributive property.

$- 2 x \cdot 15 - 2 x (- 2 x) + (27 - 2 x) (- 4 x) + (27 - 2 x) \cdot 15$

Apply the distributive property.

$- 2 x \cdot 15 - 2 x (- 2 x) + 27 (- 4 x) - 2 x (- 4 x) + (27 - 2 x) \cdot 15$

Apply the distributive property.

$- 2 x \cdot 15 - 2 x (- 2 x) + 27 (- 4 x) - 2 x (- 4 x) + 27 \cdot 15 - 2 x \cdot 15$

Combine terms.

Multiply $15$ by $- 2$ .

$- 30 x - 2 x (- 2 x) + 27 (- 4 x) - 2 x (- 4 x) + 27 \cdot 15 - 2 x \cdot 15$

Multiply $- 2$ by $- 2$ .

$- 30 x + 4 x \cdot x + 27 (- 4 x) - 2 x (- 4 x) + 27 \cdot 15 - 2 x \cdot 15$

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

$- 30 x + 4 (x^{1} x) + 27 (- 4 x) - 2 x (- 4 x) + 27 \cdot 15 - 2 x \cdot 15$

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

$- 30 x + 4 (x^{1} x^{1}) + 27 (- 4 x) - 2 x (- 4 x) + 27 \cdot 15 - 2 x \cdot 15$

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

$- 30 x + 4 x^{1 + 1} + 27 (- 4 x) - 2 x (- 4 x) + 27 \cdot 15 - 2 x \cdot 15$

Add $1$ and $1$ .

$- 30 x + 4 x^{2} + 27 (- 4 x) - 2 x (- 4 x) + 27 \cdot 15 - 2 x \cdot 15$

Multiply $- 4$ by $27$ .

$- 30 x + 4 x^{2} - 108 x - 2 x (- 4 x) + 27 \cdot 15 - 2 x \cdot 15$

Multiply $- 4$ by $- 2$ .

$- 30 x + 4 x^{2} - 108 x + 8 x \cdot x + 27 \cdot 15 - 2 x \cdot 15$

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

$- 30 x + 4 x^{2} - 108 x + 8 (x^{1} x) + 27 \cdot 15 - 2 x \cdot 15$

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

$- 30 x + 4 x^{2} - 108 x + 8 (x^{1} x^{1}) + 27 \cdot 15 - 2 x \cdot 15$

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

$- 30 x + 4 x^{2} - 108 x + 8 x^{1 + 1} + 27 \cdot 15 - 2 x \cdot 15$

Add $1$ and $1$ .

$- 30 x + 4 x^{2} - 108 x + 8 x^{2} + 27 \cdot 15 - 2 x \cdot 15$

Multiply $27$ by $15$ .

$- 30 x + 4 x^{2} - 108 x + 8 x^{2} + 405 - 2 x \cdot 15$

Multiply $15$ by $- 2$ .

$- 30 x + 4 x^{2} - 108 x + 8 x^{2} + 405 - 30 x$

Subtract $30 x$ from $- 108 x$ .

$- 30 x + 4 x^{2} - 138 x + 8 x^{2} + 405$

Subtract $138 x$ from $- 30 x$ .

$4 x^{2} - 168 x + 8 x^{2} + 405$

Add $4 x^{2}$ and $8 x^{2}$ .

$12 x^{2} - 168 x + 405$

Find the Derivative - d/dx x(15-2x)(27-2x)

Related Questions

Support LCMGCF.com by disabling your adblocker.