what is the derivative of sin(x^(1/2)) and (sin(x))^(1/2)?
Is there a difference?
Thanks!
Yes, there is a difference. Use the chain rule.
To find the derivative of sin(x^(1/2)), we can use the chain rule. The chain rule states that if we have a composition of functions, such as f(g(x)), then the derivative of that composition is given by f'(g(x)) * g'(x).
Let's break down the function sin(x^(1/2)) into two parts: the outer function f(x) = sin(x) and the inner function g(x) = x^(1/2).
To find the derivative, we need to determine the derivatives of the outer and inner functions.
The derivative of the outer function f(x) = sin(x) is simply cos(x).
The derivative of the inner function g(x) = x^(1/2) can be found using the power rule. The power rule states that the derivative of x^n is n * x^(n-1). So, applying the power rule, the derivative of g(x) = x^(1/2) is (1/2) * x^(-1/2).
Now, we can apply the chain rule. The derivative of sin(x^(1/2)) is given by f'(g(x)) * g'(x), which is cos(x^(1/2)) * (1/2) * x^(-1/2).
As for (sin(x))^(1/2), to find its derivative, we can rewrite it as sin(x)^(1/2) = (sin(x))^0.5.
Using the power rule, the derivative of (sin(x))^n is n * (sin(x))^(n-1) * cos(x), where n = 0.5.
Therefore, the derivative of (sin(x))^(1/2) is (1/2) * (sin(x))^(-1/2) * cos(x).
So, the derivatives of sin(x^(1/2)) and (sin(x))^(1/2) are cos(x^(1/2)) * (1/2) * x^(-1/2) and (1/2) * (sin(x))^(-1/2) * cos(x), respectively.