find the derivative of f(x)=x^2sec^2(3x)

also with g(x)=sqrt(cosxsecx) i get zero because cos and sec cancel each other out and derivative of a constant is zero... am i doing something wrong?

To find the derivative of the function f(x) = x^2 * sec^2(3x), we can use the product rule and the chain rule. The product rule states that if we have a function h(x) = f(x) * g(x), then the derivative of h(x), denoted as h'(x), is given by h'(x) = f'(x) * g(x) + f(x) * g'(x).

Let's break down the derivative of f(x) step by step:

Step 1: Find the derivative of x^2. Applying the power rule, we get d/dx (x^2) = 2x.

Step 2: Find the derivative of sec^2(3x). Applying the chain rule, we need to differentiate the outer function sec^2(u) and multiply it by the derivative of the inner function 3x. The derivative of sec^2(u) is d/dx(sec^2(u)) = 2sec(u) * sec(u) * tan(u). So, in our case, we have d/dx(sec^2(3x)) = 2sec(3x) * sec(3x) * tan(3x) * 3.

Step 3: Apply the product rule. Using h(x) = f(x) * g(x), where f(x) = x^2 and g(x) = sec^2(3x), we can apply the product rule as follows:
f'(x) = 2x
g'(x) = 2sec(3x) * sec(3x) * tan(3x) * 3

Now, using the product rule formula, we have:
h'(x) = f'(x) * g(x) + f(x) * g'(x)
= (2x) * sec^2(3x) + (x^2) * (2sec(3x) * sec(3x) * tan(3x) * 3)

Therefore, the derivative of f(x) = x^2 * sec^2(3x) is given by:
f'(x) = 2x * sec^2(3x) + 2x^2 * sec(3x) * sec(3x) * tan(3x) * 3

Moving on to the function g(x) = sqrt(cos(x) * sec(x)). You mentioned that you obtained a derivative of zero because the cosine and secant functions cancel each other out, and the derivative of a constant is zero. However, it seems that you misunderstood the situation.

To find the derivative of g(x), we need to apply the chain rule. The derivative will not be zero because the expression contains the product of the two functions. Here's how we can find the derivative:

Step 1: Let h(x) = cos(x) * sec(x), which is the function inside the square root.

Step 2: Find the derivative of h(x). Using the product rule, we get:
h'(x) = -sin(x) * sec(x) + cos(x) * sec(x) * tan(x)
= cos(x) * sec(x) * tan(x) - sin(x) * sec(x)

Step 3: Find the derivative of g(x) using the chain rule. Since g(x) = sqrt(h(x)), we can apply the chain rule as follows:
g'(x) = (1/2) * (h(x))^(-1/2) * h'(x)
= (1/2) * (cos(x) * sec(x))^(-1/2) * (cos(x) * sec(x) * tan(x) - sin(x) * sec(x))

Therefore, the derivative of g(x) = sqrt(cos(x) * sec(x)) is given by:
g'(x) = (1/2) * (cos(x) * sec(x))^(-1/2) * (cos(x) * sec(x) * tan(x) - sin(x) * sec(x))

So, in summary, the derivative of f(x) = x^2 * sec^2(3x) is 2x * sec^2(3x) + 2x^2 * sec(3x) * sec(3x) * tan(3x) * 3, and the derivative of g(x) = sqrt(cos(x) * sec(x)) is (1/2) * (cos(x) * sec(x))^(-1/2) * (cos(x) * sec(x) * tan(x) - sin(x) * sec(x)).