Find the derivative of cos(sqrt(e^(x^3)cos(x))

I got -sin(sqrt(e^(x^3)cos(x))*((sqrt(e^(x^3)cos(x))

Do I just leave the e^(x^3)cosx alone since it's "e" or do I still have the find the derivative for it?

The derivative of e^(x^3) has to be treated as a function (e^x) of a function (x^3). The derivative is not e^(x^3)

Thanks. I got it after 50+ tries. lol

Answer was: -sinsqrt(e^[(x^3)cos(x)])*(1/(2sqrt(e^[(x^3)cos(x)])))*e^((x^3)cos(x))*[-x^3sin(x)+3x^2cos(x)]

To find the derivative of cos(sqrt(e^(x^3)cos(x)), you can apply the chain rule.

Let's break down the function step by step:

Step 1: Identify the outer function: cos(u), where u = sqrt(e^(x^3)cos(x)).

Step 2: Differentiate the outer function cos(u) with respect to u.

The derivative of cos(u) is -sin(u).

Step 3: Take the derivative of the inner function u = sqrt(e^(x^3)cos(x)).

To differentiate this, you'll need to apply the chain rule and the product rule.

Let v = e^(x^3) and w = cos(x). Then, u = sqrt(v * w).

Now, differentiate the expression u with respect to x using the chain rule and the product rule:

du/dx = (1/2) * (v * w)^(-1/2) * (dv/dx * w + v * dw/dx)

Substituting v = e^(x^3) and w = cos(x) back in:

du/dx = (1/2) * (e^(x^3) * cos(x))^(-1/2) * (d(e^(x^3))/dx * cos(x) + e^(x^3) * (-sin(x)))

Note that we still need to differentiate the exponential term e^(x^3), as well as the cosine term cos(x).

Step 4: Differentiate the remaining terms in du/dx.

The derivative d(e^(x^3))/dx is found using the chain rule:

d(e^(x^3))/dx = 3x^2 * e^(x^3)

The derivative of cos(x) is -sin(x).

Now substitute these derivatives back into the expression for du/dx:

du/dx = (1/2) * (e^(x^3) * cos(x))^(-1/2) * (3x^2 * e^(x^3) * cos(x) + e^(x^3) * (-sin(x)))

Step 5: Simplify the expression.

To simplify further, you can factor out a common factor, e^(x^3) * cos(x), from both terms in the parentheses:

du/dx = (1/2) * (e^(x^3) * cos(x))^(-1/2) * e^(x^3) * cos(x) * (3x^2 - sin(x))

Lastly, simplify the expression inside the parentheses:

du/dx = (1/2) * (e^(x^3) * cos(x))^(-1/2) * e^(x^3) * cos(x) * (3x^2 - sin(x))

Therefore, the derivative of cos(sqrt(e^(x^3)cos(x))) is:

du/dx = (1/2) * (e^(x^3) * cos(x))^(-1/2) * e^(x^3) * cos(x) * (3x^2 - sin(x))

To find the derivative of the given function, let's break it down and apply the chain rule. The function is cos(sqrt(e^(x^3)cos(x)).

1. Let's start by assigning the inner function u = sqrt(e^(x^3)cos(x)).

2. Now, we can differentiate u with respect to x.
du/dx = d(sqrt(e^(x^3)cos(x)))/dx

3. Apply the chain rule:
du/dx = (1/2)(e^(x^3)cos(x))^(-1/2) * (d(e^(x^3)cos(x))/dx)

4. Now, let's differentiate e^(x^3)cos(x) with respect to x.
Using the product rule: (uv)' = u'v + uv', where u = e^(x^3) and v = cos(x).

du/dx = (d(e^(x^3))/dx) * cos(x) + e^(x^3) * (d(cos(x))/dx)

5. Differentiate e^(x^3) with respect to x: (d(e^(x^3))/dx = 3x^2 * e^(x^3)

6. Differentiate cos(x) with respect to x: (d(cos(x))/dx) = -sin(x)

7. Now, substitute the values back into step 4:
du/dx = (3x^2 * e^(x^3)) * cos(x) + e^(x^3) * (-sin(x))

8. Finally, substitute back u = sqrt(e^(x^3)cos(x)):
du/dx = (3x^2 * e^(x^3)) * cos(x) + e^(x^3) * (-sin(x))

So, you should keep the e^(x^3)cos(x) in the derivative and apply the chain rule to get the final result.