Differentiate the function.

f(x) = sin(x) ln(2x)

Well, let's "sin" about it for a second. To differentiate the function f(x) = sin(x) ln(2x), we'll need to use a combination of the product rule and the chain rule.

We start by using the product rule. Let's call the first term sin(x) and the second term ln(2x). The derivative of sin(x) is cos(x), and the derivative of ln(2x) is 1/(2x).

Now, applying the product rule, we have:
f'(x) = (sin(x) * (1/(2x))) + (cos(x) * ln(2x))

Now, let's simplify that a bit:
f'(x) = (sin(x)/(2x)) + (cos(x) * ln(2x))

And there you have it! The derivative of f(x) = sin(x) ln(2x) is f'(x) = (sin(x)/(2x)) + (cos(x) * ln(2x)). Just remember to keep your "sin"ning in check and you'll be all good!

To differentiate the function f(x) = sin(x) ln(2x), we will use the product rule. The product rule states that if we have two functions u(x) and v(x), the derivative of their product (u(x) * v(x)) is given by:

(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)

Let's apply the product rule to differentiate f(x) = sin(x) ln(2x):

First, we will identify u(x) and v(x):
- u(x) = sin(x)
- v(x) = ln(2x)

Next, we will find the derivatives of u(x) and v(x):
- u'(x) = cos(x) (derivative of sin(x))
- v'(x) = (1/x) * 2 (derivative of ln(2x))

Now, we can apply the product rule:

(d/dx)(f(x)) = u'(x) * v(x) + u(x) * v'(x)

= cos(x) * ln(2x) + sin(x) * (1/x) * 2

Simplifying further, we get:

= cos(x) ln(2x) + 2 sin(x) / x

Therefore, the derivative of f(x) = sin(x) ln(2x) is:

f'(x) = cos(x) ln(2x) + 2 sin(x) / x

To differentiate the function f(x) = sin(x) ln(2x), we can use the product rule and the chain rule.

The product rule states that if we have two functions u(x) and v(x), then the derivative of their product is given by:
(d/dx)(u(x) * v(x)) = u'(x) * v(x) + u(x) * v'(x)

Let's assign u(x) = sin(x) and v(x) = ln(2x).

Now, let's find the derivatives of u(x) and v(x) using the chain rule.

The derivative of sin(x) is given by:
d/dx(sin(x)) = cos(x)

The derivative of ln(2x) is given by:
d/dx(ln(2x)) = 1/(2x) * d/dx(2x)
= 1/(2x) * 2
= 1/x

Now we can find the derivative of f(x) by applying the product rule:

f'(x) = u'(x) * v(x) + u(x) * v'(x)
= cos(x) * ln(2x) + sin(x) * (1/x)
= cos(x)ln(2x) + sin(x)/x

So, the derivative of f(x) = sin(x) ln(2x) is f'(x) = cos(x)ln(2x) + sin(x)/x.

just use the product rule:

f = sin(x) ln(2x)
f' = cos(x) ln(2x) + sin(x) * 1/(2x) * 2
= cos(x) ln(2x) + sin(x)/x