How do you find #(d^2y)/(dx^2)# given #x=tany#?
By signing up, you agree to our Terms of Service and Privacy Policy
To differentiate without using the inverse tangent, see below.
To see that this is the same as the other answer
Now,
By signing up, you agree to our Terms of Service and Privacy Policy
To find (d^2y)/(dx^2) given x = tan y, we'll start by expressing y in terms of x using the inverse tangent function. Then, we'll differentiate twice with respect to x.
Given x = tan y, we can solve for y: y = arctan(x)
Now, we differentiate y with respect to x: (dy/dx) = d(arctan(x))/dx Using the chain rule, this becomes: (dy/dx) = (1 / (1 + x^2)) * dx/dy
To find (dx/dy), we take the reciprocal of the derivative of x with respect to y: (dx/dy) = 1 / (dy/dx)
Now, we differentiate (dy/dx) with respect to x to find (d^2y)/(dx^2): (d^2y)/(dx^2) = d((dy/dx))/dx Using the chain rule again, this becomes: (d^2y)/(dx^2) = d((dy/dx)/(dx/dy))/dx
Substituting (dy/dx) and (dx/dy) into the equation, we get: (d^2y)/(dx^2) = d((1 / (1 + x^2)) * (dy/dx))/dx
Now, we differentiate this expression with respect to x: (d^2y)/(dx^2) = d((1 / (1 + x^2)) * (1 / (dy/dx)))/dx
Solving this derivative will give us the second derivative (d^2y)/(dx^2) in terms of x.
Therefore, the expression for (d^2y)/(dx^2) given x = tan y is obtained by differentiating the expression (1 / (1 + x^2)) * (1 / (dy/dx)) twice with respect to x.
By signing up, you agree to our Terms of Service and Privacy Policy
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7