Using the chain rule, find the first derivative of y=tan^2(3x)

Answer 1

To find the derivative of y = tan^2(3x), we need to use the chain rule.

Let u = 3x. Then y = tan^2(u).

Now, let's find the derivative of y with respect to u:

dy/du = d/dy (tan^2(u))

To find this derivative, we can use the chain rule:

dy/du = d/dy (tan(u))^2 * d/du (u)

The derivative of tan(u)^2 with respect to tan(u) is 2tan(u), and the derivative of u with respect to x (du/dx) is 3:

dy/du = 2tan(u) * 3

Now, let's substitute u back in:

dy/du = 2tan(3x)

Therefore, the first derivative of y = tan^2(3x) with respect to x is 2tan(3x).

Answer 2

AAAaannndd the bot gets it wrong yet again!
all that work, and you forgot the chain rule.

dy/dx = 2tan(3x) * sec^2(3x) * 3 = 6 tan(3x) sec^2(3x)

Answer 3

Apologies for the previous incorrect response. You are absolutely correct. Let's use the chain rule to find the first derivative of y = tan^2(3x).

Let u = 3x. Then y = tan^2(u).

We can now find the derivative of y with respect to x:

dy/dx = (d/dx) (tan^2(u))

To find this derivative, we can use the chain rule:

dy/dx = (d/du) (tan^2(u)) * (du/dx)

We know that (d/du) (tan^2(u)) = 2tan(u) * (sec(u))^2 and (du/dx) = 3:

dy/dx = 2tan(u) * (sec(u))^2 * 3

Now, let's substitute u back in:

dy/dx = 6tan(3x) * (sec(3x))^2

Therefore, the correct first derivative of y = tan^2(3x) with respect to x is dy/dx = 6tan(3x) * (sec(3x))^2.