what are the first and second derivatives for y= x* (4-x^2)^1/2
To find the first and second derivatives of the function y = x(4 - x^2)^(1/2), we can use the chain rule and the power rule.
Step 1: Let's rewrite the function in a different form. First, we can express (4 - x^2)^(1/2) as (4 - x^2)^(1/2) * x^0. Now, our function is y = x^(1)*(4 - x^2)^(1/2).
Step 2: Applying the chain rule, we differentiate the function with respect to x. The chain rule states that if we have y = f(g(x)), where f(u) and g(x) are differentiable functions, then the derivative of y with respect to x is given by dy/dx = f'(g(x)) * g'(x).
In our case, f(u) = u^(1/2) and g(x) = 4 - x^2.
So, using the chain rule, we have dy/dx = (4 - x^2)^(1/2) * d/dx(x^1) + x(1/2)*(4 - x^2)^(-1/2) * d/dx(4 - x^2).
Simplifying, we get dy/dx = (4 - x^2)^(1/2) + x(1/2)*(4 - x^2)^(-1/2)* (-2x).
Step 3: To find the second derivative, we differentiate again using the same rules.
For dy/dx = (4 - x^2)^(1/2) - x^2/(4 - x^2)^(1/2), we use the quotient rule. The quotient rule states that if we have y = f(x)/g(x), where f(x) and g(x) are differentiable functions, then the derivative of y with respect to x is given by dy/dx = (f'(x) * g(x) - f(x) * g'(x)) / (g(x)^2).
Differentiating, we get dy/dx = (-x^2 * (-1/(4 - x^2)^(1/2)) + (4 - x^2)^(1/2) * 2x) / (4 - x^2).
Simplifying further, we obtain dy/dx = (2x^3 - 4x) / ((4 - x^2)^(3/2)).
Therefore, the first derivative of y = x(4 - x^2)^(1/2) is dy/dx = (4 - x^2)^(1/2) + x(1/2)*(4 - x^2)^(-1/2)* (-2x), and the second derivative is d^2y/dx^2 = (2x^3 - 4x) / ((4 - x^2)^(3/2)).