Using the chain rule, find the first derivative of y=tan^2(3x)
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)
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.
To find the derivative of y = tan^2(3x) using the chain rule, follow these steps:
Step 1: Rewrite the function using exponent notation:
y = (tan(3x))^2
Step 2: Let u = 3x, then rewrite the function in terms of u:
y = (tan(u))^2
Step 3: Find the derivative of the function with respect to u:
dy/du = 2tan(u) * sec^2(u)
Step 4: Substitute back u = 3x to get the derivative in terms of x:
dy/dx = dy/du * du/dx = 2tan(u) * sec^2(u) * 3
Step 5: Simplify the expression:
dy/dx = 6tan(3x) * sec^2(3x)
So, the first derivative of y = tan^2(3x) is dy/dx = 6tan(3x) * sec^2(3x).
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).