How do I find the derivative for the sqrt(sin(e^(x^3)*cos(x)))???

I know this is a combination of the chain rule and product rule. Please help!

Nothing?

You know, if no one responds right away, a better follow-up would be to show whatever ideas you have had in the meantime, rather than just tapping your foot and saying "nothing?".

Your first thought, on seeing a complicated formula should have been the chain rule. what functions u and v could you use?

y = √u where
u = sin(e^(x^3)*cos(x))

u = sin v
where v = e^(x^3)*cos(x)

Now you can work with things:

y' = 1/2√u u'
= 1/2√u cos v v'
Now use the product rule:
= 1/2√u cos v e^(x^3) (3x^2 cosx - x^3 sinx)

Now just substitute back in for u and v.

Thanks for your response. I was hardly taping my foot. I have been working on this problem for almost 6 days and by the time I posted this, I was over it. So I apologize if it seemed like I was impatient.

You are the man!!!! I had tried this problem freaking 74 times. I had too many parenthesis, I forgot to add (e^(x^3)), and finally at the end of the derivative I did NOT include X^3 in front of sin(x).

Thank youuuuuuuu

Good work. Many many times it is easier to make a few simple substitutions to avoid all the punctuation. Even the most complex problems contain simple parts.

To find the derivative of a function like sqrt(sin(e^(x^3) * cos(x))), we need to apply the chain rule and the product rule.

Let's break it down step by step:

Step 1: Identify the functions and their derivatives.

The outermost function is the square root function, f(x) = sqrt(g(x)), where g(x) = sin(e^(x^3) * cos(x)). The derivative of the square root function is 1 / (2 * sqrt(g(x))).

Step 2: Apply the chain rule.

To apply the chain rule, we need to find the derivative of the innermost function and then multiply it by the derivative of the outermost function.

Let's start with the innermost function g(x) = sin(e^(x^3) * cos(x)).

The derivative of sin(x) is cos(x), and the derivative of cos(x) is -sin(x). So, the derivative of g(x) is:

g'(x) = cos(e^(x^3) * cos(x)) * (e^(x^3) * -sin(x)).

Step 3: Apply the product rule.

Now that we have the derivative of the innermost function, we can use the product rule to find the derivative of f(x) = sqrt(g(x)).

The product rule states that if we have two functions, u(x) and v(x), then the derivative of their product is given by:

(f * g)' = u' * v + u * v',

where u' represents the derivative of u and v' represents the derivative of v.

In our case, u(x) = 1 / (2 * sqrt(g(x))), v(x) = g(x), and g'(x) = cos(e^(x^3) * cos(x)) * (e^(x^3) * -sin(x)).

Applying the product rule, we have:

f'(x) = [u' * v]' + u * v'
= [(-1 / (4 * (g(x))^(3/2))) * cos(e^(x^3) * cos(x)) * (e^(x^3)* -sin(x))] + (1 / (2 * sqrt(g(x)))) * cos(e^(x^3) * cos(x)) * (e^(x^3) * -sin(x)).

Simplifying this expression will give the final derivative of the given function.

This process might seem somewhat complex, but by breaking down the function and applying the chain rule and product rule, we can find the derivative step by step. Remember to double-check your work and simplify the answer where possible. You can also make use of software like Wolfram Alpha or symbolic calculators for complex derivatives.