Find the Derivative - d/dx 5x^2(4+x)

$5 x^{2} (4 + x)$

Step 1

Since

$5$ is constant with respect to

$x$ , the derivative of

$5 x^{2} (4 + x)$ with respect to

$x$ is

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

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

Step 2

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

$f (x) = x^{2}$ and

$g (x) = 4 + x$ .

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

Step 3

Differentiate.

By the Sum Rule, the derivative of

$4 + x$ with respect to

$x$ is

$\frac{d}{d x} [4] + \frac{d}{d x} [x]$ .

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

Since

$4$ is constant with respect to

$x$ , the derivative of

$4$ with respect to

$x$ is

$0$ .

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

Add

$0$ and

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

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

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 1$ .

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

Multiply

$x^{2}$ by

$1$ .

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

Differentiate using the Power Rule which states that

$\frac{d}{d x} [x^{n}]$ is

$n x^{n - 1}$ where

$n = 2$ .

$5 (x^{2} + (4 + x) (2 x))$

Move

$2$ to the left of

$4 + x$ .

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

Step 4

Simplify.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Combine terms.

Multiply

$2$ by

$4$ .

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

Multiply

$8$ by

$5$ .

$5 x^{2} + 40 x + 5 (2 x \cdot x)$

Raise

$x$ to the power of

$1$ .

$5 x^{2} + 40 x + 5 (2 (x^{1} x))$

Raise

$x$ to the power of

$1$ .

$5 x^{2} + 40 x + 5 (2 (x^{1} x^{1}))$

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$5 x^{2} + 40 x + 5 (2 x^{1 + 1})$

Add

$1$ and

$1$ .

$5 x^{2} + 40 x + 5 (2 x^{2})$

Multiply

$2$ by

$5$ .

$5 x^{2} + 40 x + 10 x^{2}$

Add

$5 x^{2}$ and

$10 x^{2}$ .

$15 x^{2} + 40 x$

Find the Derivative - d/dx 5x^2(4+x)

Related Questions