Find the Derivative Using Product Rule - d/d@VAR f(x)=(x^2-4x+2)(2x^3-x^2+4)

$f (x) = (x^{2} - 4 x + 2) (2 x^{3} - x^{2} + 4)$

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 $f (x) = x^{2} - 4 x + 2$ and $g (x) = 2 x^{3} - x^{2} + 4$ .

$(x^{2} - 4 x + 2) \frac{d}{d x} [2 x^{3} - x^{2} + 4] + (2 x^{3} - x^{2} + 4) \frac{d}{d x} [x^{2} - 4 x + 2]$

Step 2

By the Sum Rule, the derivative of $2 x^{3} - x^{2} + 4$ with respect to $x$ is $\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [4]$ .

$(x^{2} - 4 x + 2) (\frac{d}{d x} [2 x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [4]) + (2 x^{3} - x^{2} + 4) \frac{d}{d x} [x^{2} - 4 x + 2]$

Step 3

Evaluate $\frac{d}{d x} [2 x^{3}]$ .

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

$(x^{2} - 4 x + 2) (2 \frac{d}{d x} [x^{3}] + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [4]) + (2 x^{3} - x^{2} + 4) \frac{d}{d x} [x^{2} - 4 x + 2]$

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$(x^{2} - 4 x + 2) (2 (3 x^{2}) + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [4]) + (2 x^{3} - x^{2} + 4) \frac{d}{d x} [x^{2} - 4 x + 2]$

Multiply $3$ by $2$ .

$(x^{2} - 4 x + 2) (6 x^{2} + \frac{d}{d x} [- x^{2}] + \frac{d}{d x} [4]) + (2 x^{3} - x^{2} + 4) \frac{d}{d x} [x^{2} - 4 x + 2]$

Step 4

Evaluate $\frac{d}{d x} [- x^{2}]$ .

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$(x^{2} - 4 x + 2) (6 x^{2} - \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4]) + (2 x^{3} - x^{2} + 4) \frac{d}{d x} [x^{2} - 4 x + 2]$

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

$(x^{2} - 4 x + 2) (6 x^{2} - (2 x) + \frac{d}{d x} [4]) + (2 x^{3} - x^{2} + 4) \frac{d}{d x} [x^{2} - 4 x + 2]$

Multiply $2$ by $- 1$ .

$(x^{2} - 4 x + 2) (6 x^{2} - 2 x + \frac{d}{d x} [4]) + (2 x^{3} - x^{2} + 4) \frac{d}{d x} [x^{2} - 4 x + 2]$

Step 5

Differentiate.

Since $4$ is constant with respect to $x$ , the derivative of $4$ with respect to $x$ is $0$ .

$(x^{2} - 4 x + 2) (6 x^{2} - 2 x + 0) + (2 x^{3} - x^{2} + 4) \frac{d}{d x} [x^{2} - 4 x + 2]$

Add $6 x^{2} - 2 x$ and $0$ .

$(x^{2} - 4 x + 2) (6 x^{2} - 2 x) + (2 x^{3} - x^{2} + 4) \frac{d}{d x} [x^{2} - 4 x + 2]$

By the Sum Rule, the derivative of $x^{2} - 4 x + 2$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2]$ .

$(x^{2} - 4 x + 2) (6 x^{2} - 2 x) + (2 x^{3} - x^{2} + 4) (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2])$

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

$(x^{2} - 4 x + 2) (6 x^{2} - 2 x) + (2 x^{3} - x^{2} + 4) (2 x + \frac{d}{d x} [- 4 x] + \frac{d}{d x} [2])$

Step 6

Evaluate $\frac{d}{d x} [- 4 x]$ .

Since $- 4$ is constant with respect to $x$ , the derivative of $- 4 x$ with respect to $x$ is $- 4 \frac{d}{d x} [x]$ .

$(x^{2} - 4 x + 2) (6 x^{2} - 2 x) + (2 x^{3} - x^{2} + 4) (2 x - 4 \frac{d}{d x} [x] + \frac{d}{d x} [2])$

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

$(x^{2} - 4 x + 2) (6 x^{2} - 2 x) + (2 x^{3} - x^{2} + 4) (2 x - 4 \cdot 1 + \frac{d}{d x} [2])$

Multiply $- 4$ by $1$ .

$(x^{2} - 4 x + 2) (6 x^{2} - 2 x) + (2 x^{3} - x^{2} + 4) (2 x - 4 + \frac{d}{d x} [2])$

Step 7

Differentiate using the Constant Rule.

Since $2$ is constant with respect to $x$ , the derivative of $2$ with respect to $x$ is $0$ .

$(x^{2} - 4 x + 2) (6 x^{2} - 2 x) + (2 x^{3} - x^{2} + 4) (2 x - 4 + 0)$

Add $2 x - 4$ and $0$ .

$(x^{2} - 4 x + 2) (6 x^{2} - 2 x) + (2 x^{3} - x^{2} + 4) (2 x - 4)$

Step 8

Simplify.

Factor $1$ out of $(x^{2} - 4 x + 2) (6 x^{2} - 2 x) + (2 x^{3} - x^{2} + 4) (2 x - 4)$ .

Factor $1$ out of $(x^{2} - 4 x + 2) (6 x^{2} - 2 x)$ .

$1 ((x^{2} - 4 x + 2) (6 x^{2} - 2 x)) + (2 x^{3} - x^{2} + 4) (2 x - 4)$

Factor $1$ out of $(2 x^{3} - x^{2} + 4) (2 x - 4)$ .

$1 ((x^{2} - 4 x + 2) (6 x^{2} - 2 x)) + 1 ((2 x^{3} - x^{2} + 4) (2 x - 4))$

Factor $1$ out of $1 ((x^{2} - 4 x + 2) (6 x^{2} - 2 x)) + 1 ((2 x^{3} - x^{2} + 4) (2 x - 4))$ .

$1 ((x^{2} - 4 x + 2) (6 x^{2} - 2 x) + (2 x^{3} - x^{2} + 4) (2 x - 4))$

Multiply $(x^{2} - 4 x + 2) (6 x^{2} - 2 x) + (2 x^{3} - x^{2} + 4) (2 x - 4)$ by $1$ .

$(x^{2} - 4 x + 2) (6 x^{2} - 2 x) + (2 x^{3} - x^{2} + 4) (2 x - 4)$

Reorder terms.

$(6 x^{2} - 2 x) (x^{2} - 4 x + 2) + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Simplify each term.

Expand $(6 x^{2} - 2 x) (x^{2} - 4 x + 2)$ by multiplying each term in the first expression by each term in the second expression.

$6 x^{2} x^{2} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 2 - 2 x \cdot x^{2} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Simplify each term.

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

Move $x^{2}$ .

$6 (x^{2} x^{2}) + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 2 - 2 x \cdot x^{2} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

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

$6 x^{2 + 2} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 2 - 2 x \cdot x^{2} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Add $2$ and $2$ .

$6 x^{4} + 6 x^{2} (- 4 x) + 6 x^{2} \cdot 2 - 2 x \cdot x^{2} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Rewrite using the commutative property of multiplication.

$6 x^{4} + 6 \cdot - 4 x^{2} x + 6 x^{2} \cdot 2 - 2 x \cdot x^{2} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

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

Move $x$ .

$6 x^{4} + 6 \cdot - 4 (x \cdot x^{2}) + 6 x^{2} \cdot 2 - 2 x \cdot x^{2} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Multiply $x$ by $x^{2}$ .

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

$6 x^{4} + 6 \cdot - 4 (x^{1} x^{2}) + 6 x^{2} \cdot 2 - 2 x \cdot x^{2} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

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

$6 x^{4} + 6 \cdot - 4 x^{1 + 2} + 6 x^{2} \cdot 2 - 2 x \cdot x^{2} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Add $1$ and $2$ .

$6 x^{4} + 6 \cdot - 4 x^{3} + 6 x^{2} \cdot 2 - 2 x \cdot x^{2} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Multiply $6$ by $- 4$ .

$6 x^{4} - 24 x^{3} + 6 x^{2} \cdot 2 - 2 x \cdot x^{2} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Multiply $2$ by $6$ .

$6 x^{4} - 24 x^{3} + 12 x^{2} - 2 x \cdot x^{2} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

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

Move $x^{2}$ .

$6 x^{4} - 24 x^{3} + 12 x^{2} - 2 (x^{2} x) - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Multiply $x^{2}$ by $x$ .

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

$6 x^{4} - 24 x^{3} + 12 x^{2} - 2 (x^{2} x^{1}) - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

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

$6 x^{4} - 24 x^{3} + 12 x^{2} - 2 x^{2 + 1} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Add $2$ and $1$ .

$6 x^{4} - 24 x^{3} + 12 x^{2} - 2 x^{3} - 2 x (- 4 x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Rewrite using the commutative property of multiplication.

$6 x^{4} - 24 x^{3} + 12 x^{2} - 2 x^{3} - 2 \cdot - 4 x \cdot x - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$6 x^{4} - 24 x^{3} + 12 x^{2} - 2 x^{3} - 2 \cdot - 4 (x \cdot x) - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Multiply $x$ by $x$ .

$6 x^{4} - 24 x^{3} + 12 x^{2} - 2 x^{3} - 2 \cdot - 4 x^{2} - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Multiply $- 2$ by $- 4$ .

$6 x^{4} - 24 x^{3} + 12 x^{2} - 2 x^{3} + 8 x^{2} - 2 x \cdot 2 + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Multiply $2$ by $- 2$ .

$6 x^{4} - 24 x^{3} + 12 x^{2} - 2 x^{3} + 8 x^{2} - 4 x + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Subtract $2 x^{3}$ from $- 24 x^{3}$ .

$6 x^{4} - 26 x^{3} + 12 x^{2} + 8 x^{2} - 4 x + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Add $12 x^{2}$ and $8 x^{2}$ .

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + (2 x - 4) (2 x^{3} - x^{2} + 4)$

Expand $(2 x - 4) (2 x^{3} - x^{2} + 4)$ by multiplying each term in the first expression by each term in the second expression.

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 2 x (2 x^{3}) + 2 x (- x^{2}) + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

Simplify each term.

Rewrite using the commutative property of multiplication.

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 2 \cdot 2 x \cdot x^{3} + 2 x (- x^{2}) + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

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

Move $x^{3}$ .

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 2 \cdot 2 (x^{3} x) + 2 x (- x^{2}) + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

Multiply $x^{3}$ by $x$ .

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

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 2 \cdot 2 (x^{3} x^{1}) + 2 x (- x^{2}) + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

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

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 2 \cdot 2 x^{3 + 1} + 2 x (- x^{2}) + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

Add $3$ and $1$ .

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 2 \cdot 2 x^{4} + 2 x (- x^{2}) + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

Multiply $2$ by $2$ .

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 4 x^{4} + 2 x (- x^{2}) + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

Rewrite using the commutative property of multiplication.

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 4 x^{4} + 2 \cdot - 1 x \cdot x^{2} + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

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

Move $x^{2}$ .

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 4 x^{4} + 2 \cdot - 1 (x^{2} x) + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

Multiply $x^{2}$ by $x$ .

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

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 4 x^{4} + 2 \cdot - 1 (x^{2} x^{1}) + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

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

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 4 x^{4} + 2 \cdot - 1 x^{2 + 1} + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

Add $2$ and $1$ .

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 4 x^{4} + 2 \cdot - 1 x^{3} + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

Multiply $2$ by $- 1$ .

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 4 x^{4} - 2 x^{3} + 2 x \cdot 4 - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

Multiply $4$ by $2$ .

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 4 x^{4} - 2 x^{3} + 8 x - 4 (2 x^{3}) - 4 (- x^{2}) - 4 \cdot 4$

Multiply $2$ by $- 4$ .

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 4 x^{4} - 2 x^{3} + 8 x - 8 x^{3} - 4 (- x^{2}) - 4 \cdot 4$

Multiply $- 1$ by $- 4$ .

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 4 x^{4} - 2 x^{3} + 8 x - 8 x^{3} + 4 x^{2} - 4 \cdot 4$

Multiply $- 4$ by $4$ .

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 4 x^{4} - 2 x^{3} + 8 x - 8 x^{3} + 4 x^{2} - 16$

Subtract $8 x^{3}$ from $- 2 x^{3}$ .

$6 x^{4} - 26 x^{3} + 20 x^{2} - 4 x + 4 x^{4} - 10 x^{3} + 8 x + 4 x^{2} - 16$

Add $6 x^{4}$ and $4 x^{4}$ .

$10 x^{4} - 26 x^{3} + 20 x^{2} - 4 x - 10 x^{3} + 8 x + 4 x^{2} - 16$

Subtract $10 x^{3}$ from $- 26 x^{3}$ .

$10 x^{4} - 36 x^{3} + 20 x^{2} - 4 x + 8 x + 4 x^{2} - 16$

Add $20 x^{2}$ and $4 x^{2}$ .

$10 x^{4} - 36 x^{3} + 24 x^{2} - 4 x + 8 x - 16$

Add $- 4 x$ and $8 x$ .

$10 x^{4} - 36 x^{3} + 24 x^{2} + 4 x - 16$

Find the Derivative Using Product Rule - d/d@VAR f(x)=(x^2-4x+2)(2x^3-x^2+4)

Related Questions

Support LCMGCF.com by disabling your adblocker.