Find the Derivative Using Product Rule - d/dx y=(3x^2+4)(2x-5)

$y = (3 x^{2} + 4) (2 x - 5)$

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 $f (x) = 3 x^{2} + 4$ and $g (x) = 2 x - 5$ .

$(3 x^{2} + 4) \frac{d}{d x} [2 x - 5] + (2 x - 5) \frac{d}{d x} [3 x^{2} + 4]$

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

$(3 x^{2} + 4) (\frac{d}{d x} [2 x] + \frac{d}{d x} [- 5]) + (2 x - 5) \frac{d}{d x} [3 x^{2} + 4]$

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

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

$(3 x^{2} + 4) (2 \frac{d}{d x} [x] + \frac{d}{d x} [- 5]) + (2 x - 5) \frac{d}{d x} [3 x^{2} + 4]$

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

$(3 x^{2} + 4) (2 \cdot 1 + \frac{d}{d x} [- 5]) + (2 x - 5) \frac{d}{d x} [3 x^{2} + 4]$

Multiply $2$ by $1$ .

$(3 x^{2} + 4) (2 + \frac{d}{d x} [- 5]) + (2 x - 5) \frac{d}{d x} [3 x^{2} + 4]$

Differentiate.

Since $- 5$ is constant with respect to $x$ , the derivative of $- 5$ with respect to $x$ is $0$ .

$(3 x^{2} + 4) (2 + 0) + (2 x - 5) \frac{d}{d x} [3 x^{2} + 4]$

Add $2$ and $0$ .

$(3 x^{2} + 4) \cdot 2 + (2 x - 5) \frac{d}{d x} [3 x^{2} + 4]$

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

$(3 x^{2} + 4) \cdot 2 + (2 x - 5) (\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [4])$

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

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

$(3 x^{2} + 4) \cdot 2 + (2 x - 5) (3 \frac{d}{d x} [x^{2}] + \frac{d}{d x} [4])$

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

$(3 x^{2} + 4) \cdot 2 + (2 x - 5) (3 (2 x) + \frac{d}{d x} [4])$

Multiply $2$ by $3$ .

$(3 x^{2} + 4) \cdot 2 + (2 x - 5) (6 x + \frac{d}{d x} [4])$

Differentiate using the Constant Rule.

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

$(3 x^{2} + 4) \cdot 2 + (2 x - 5) (6 x + 0)$

Add $6 x$ and $0$ .

$(3 x^{2} + 4) \cdot 2 + (2 x - 5) (6 x)$

Simplify.

Apply the distributive property.

$3 x^{2} \cdot 2 + 4 \cdot 2 + (2 x - 5) (6 x)$

Apply the distributive property.

$3 x^{2} \cdot 2 + 4 \cdot 2 + 2 x (6 x) - 5 (6 x)$

Combine terms.

Multiply $2$ by $3$ .

$6 x^{2} + 4 \cdot 2 + 2 x (6 x) - 5 (6 x)$

Multiply $4$ by $2$ .

$6 x^{2} + 8 + 2 x (6 x) - 5 (6 x)$

Multiply $6$ by $2$ .

$6 x^{2} + 8 + 12 x \cdot x - 5 (6 x)$

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

$6 x^{2} + 8 + 12 (x^{1} x) - 5 (6 x)$

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

$6 x^{2} + 8 + 12 (x^{1} x^{1}) - 5 (6 x)$

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

$6 x^{2} + 8 + 12 x^{1 + 1} - 5 (6 x)$

Add $1$ and $1$ .

$6 x^{2} + 8 + 12 x^{2} - 5 (6 x)$

Multiply $6$ by $- 5$ .

$6 x^{2} + 8 + 12 x^{2} - 30 x$

Add $6 x^{2}$ and $12 x^{2}$ .

$18 x^{2} + 8 - 30 x$

Reorder terms.

$18 x^{2} - 30 x + 8$

Find the Derivative Using Product Rule - d/dx y=(3x^2+4)(2x-5)

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