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

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

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

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

Step 2

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

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

Step 3

Evaluate $\frac{d}{d x} [5 x]$ .

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

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

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

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

Multiply $5$ by $1$ .

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

Step 4

Differentiate.

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

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

Add $5$ and $0$ .

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

By the Sum Rule, the derivative of $3 x^{2} + 5$ with respect to $x$ is $\frac{d}{d x} [3 x^{2}] + \frac{d}{d x} [5]$ .

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

Step 5

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

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

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

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

Multiply $2$ by $3$ .

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

Step 6

Differentiate using the Constant Rule.

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

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

Add $6 x$ and $0$ .

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

Step 7

Simplify.

Apply the distributive property.

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

Apply the distributive property.

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

Combine terms.

Multiply $5$ by $3$ .

$15 x^{2} + 5 \cdot 5 + 5 x (6 x) - 2 (6 x)$

Multiply $5$ by $5$ .

$15 x^{2} + 25 + 5 x (6 x) - 2 (6 x)$

Multiply $6$ by $5$ .

$15 x^{2} + 25 + 30 x \cdot x - 2 (6 x)$

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

$15 x^{2} + 25 + 30 (x^{1} x) - 2 (6 x)$

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

$15 x^{2} + 25 + 30 (x^{1} x^{1}) - 2 (6 x)$

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

$15 x^{2} + 25 + 30 x^{1 + 1} - 2 (6 x)$

Add $1$ and $1$ .

$15 x^{2} + 25 + 30 x^{2} - 2 (6 x)$

Multiply $6$ by $- 2$ .

$15 x^{2} + 25 + 30 x^{2} - 12 x$

Add $15 x^{2}$ and $30 x^{2}$ .

$45 x^{2} + 25 - 12 x$

Reorder terms.

$45 x^{2} - 12 x + 25$

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

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