Let
y = tan^(-1)(sqrt(5x^(2)-1))
Find dy/dx
To find dy/dx, we need to take the derivative of y with respect to x. Let's break down the function and use the chain rule to find the derivative.
Given: y = tan^(-1)(sqrt(5x^(2)-1))
Step 1: Rewrite the expression using trigonometric and algebraic identities.
y = arctan(sqrt(5x^(2)-1))
Step 2: Apply the chain rule.
The derivative of y with respect to x can be expressed as:
dy/dx = d(arctan(u))/du * du/dx
where u = sqrt(5x^(2)-1).
Step 3: Compute the derivatives.
The derivative of arctan(u) with respect to u is 1/(1+u^(2)).
The derivative of sqrt(5x^(2)-1) with respect to x can be found by applying the chain rule, which gives us (1/2)*(5x^(2)-1)^(-1/2)*10x.
Step 4: Substitute the computed derivatives back into the chain rule equation.
dy/dx = (1/(1+u^(2))) * (1/2)*(5x^(2)-1)^(-1/2)*10x
dy/dx = (10x)/(2(1+5x^(2)-1))
Simplifying further,
dy/dx = (5x)/(1+5x^(2))
Therefore, the derivative dy/dx is (5x)/(1+5x^(2)).