find the derivative y= csc^-1 (secx)
0 less than or equal to x less than or equal to 2¤Ð
what did i do wrong?
- 1/ secx sqrt(secx^2 -1) * secxtanx
Oh no, it seems like you missed a small detail there! When finding the derivative of y = csc^(-1)(secx), you forgot to apply the chain rule correctly. Don't worry, everyone makes mistakes!
To fix it, let's start from the beginning. The derivative of y = csc^(-1)(secx) can be found using the chain rule.
First, let's rewrite y = csc^(-1)(secx) as y = arcsin(1/secx).
Now, the derivative can be found as follows:
dy/dx = (d(arcsin(u))/du) * (du/dx)
In this case, u = 1/secx and we need to find du/dx. The derivative of 1/secx is -secxtanx.
So now, we have:
dy/dx = (d(arcsin(u))/du) * (-secxtanx)
Now, let's find d(arcsin(u))/du. The derivative of arcsin(u) is 1/sqrt(1 - u^2). In this case, u = 1/secx, so we have:
d(arcsin(u))/du = 1/sqrt(1 - (1/secx)^2) = 1/sqrt(1 - 1/(secx)^2) = 1/sqrt(secx^2 - 1)
Substituting this back into the derivative expression, we get:
dy/dx = (1/sqrt(secx^2 - 1)) * (-secxtanx)
So, the correct derivative is:
dy/dx = -secxtanx / sqrt(secx^2 - 1)
Hope this clears things up! Keep smiling and learning!
To find the derivative of y = csc^-1(secx), you need to use the chain rule. Here is the correct step-by-step solution:
Step 1: Start by rewriting y = csc^-1(secx) as y = arcsin(1/secx).
Step 2: Apply the chain rule:
dy/dx = (d(arcsin(u))/du) * (du/dx),
where u = 1/secx.
Step 3: Find the derivatives:
The derivative of arcsin(u) with respect to u is 1/sqrt(1-u^2).
The derivative of u = 1/secx with respect to x is -secxtanx.
Step 4: Plug in the derivatives into the chain rule:
dy/dx = (1/sqrt(1-(1/secx)^2)) * (-secxtanx).
Step 5: Simplify the expression:
dy/dx = -secxtanx / sqrt((secx)^2 - 1).
Therefore, the correct derivative is -secxtanx / sqrt((secx)^2 - 1).
It seems like you made a mistake in the differentiation step by not considering the negative sign in front of secxtanx.
To find the derivative of y = csc^-1(secx), you can use the chain rule.
First, let's rewrite the given function as y = arcsin(1/secx).
To find the derivative, you need to understand that the derivative of arcsin(u) with respect to u is 1/sqrt(1 - u^2).
Now, let u = 1/secx.
To find the derivative of u, use the chain rule. The derivative of u with respect to x is:
du/dx = (-1/secx^2) * secxtanx
Now, substitute u and du/dx back into the derivative of y with respect to x:
dy/dx = (1/sqrt(1 - u^2)) * (du/dx)
dy/dx = (1/sqrt(1 - (1/secx)^2)) * ((-1/secx^2) * secxtanx)
Simplify further:
dy/dx = -secxtanx/(sqrt(secx^2 - 1) * secx^2)
So, the correct derivative is -secxtanx/(sqrt(secx^2 - 1) * secx^2).
If you believe you made an error in your work, double-check your calculations and make sure you applied the chain rule correctly.