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

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

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

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

Step 2

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

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

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

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

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

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

Multiply $5$ by $1$ .

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

Step 4

Differentiate.

Since $- 3$ is constant with respect to $x$ , the derivative of $- 3$ with respect to $x$ is $0$ .

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

Add $5$ and $0$ .

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

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

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

Step 5

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

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

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

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

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

Multiply $2$ by $2$ .

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

Step 6

Differentiate using the Constant Rule.

Since $3$ is constant with respect to $x$ , the derivative of $3$ with respect to $x$ is $0$ .

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

Add $4 x$ and $0$ .

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

Step 7

Simplify.

Apply the distributive property.

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

Apply the distributive property.

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

Combine terms.

Multiply $5$ by $2$ .

$10 x^{2} + 3 \cdot 5 + 5 x (4 x) - 3 (4 x)$

Multiply $3$ by $5$ .

$10 x^{2} + 15 + 5 x (4 x) - 3 (4 x)$

Multiply $4$ by $5$ .

$10 x^{2} + 15 + 20 x \cdot x - 3 (4 x)$

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

$10 x^{2} + 15 + 20 (x^{1} x) - 3 (4 x)$

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

$10 x^{2} + 15 + 20 (x^{1} x^{1}) - 3 (4 x)$

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

$10 x^{2} + 15 + 20 x^{1 + 1} - 3 (4 x)$

Add $1$ and $1$ .

$10 x^{2} + 15 + 20 x^{2} - 3 (4 x)$

Multiply $4$ by $- 3$ .

$10 x^{2} + 15 + 20 x^{2} - 12 x$

Add $10 x^{2}$ and $20 x^{2}$ .

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

Reorder terms.

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

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

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