Determine dy/dx if:
(i) Y= sqrt x^sqrt x
(ii) Y= xarcsin(x) + sqrt(1-x)
Please show all steps. Thanks
1) http://www.analyzemath.com/calculus/Differentiation/first_derivative.html
2) an antiderivative of arcsin(x) is xarcsin(x)-sqrt(1-x^2); are you certain the second term is not sqrt(1-x^2)?
To find the derivative of a function, we can use the rules of differentiation.
For the first problem:
(i) Y = sqrt(x^sqrt(x))
To find dy/dx, we will use the chain rule.
Let u = x^sqrt(x).
Then, Y = sqrt(u).
Now let's find du/dx:
Using the power rule, we have:
du/dx = d/dx (x^sqrt(x))
= sqrt(x) * d/dx (x^sqrt(x-1))
= sqrt(x) * sqrt(x-1) * d/dx (x-1)
Using the power rule again, we have:
du/dx = sqrt(x) * sqrt(x-1) * (x-1)^(sqrt(x)-1) * 1
Now, let's find dy/du:
Using the power rule, we have:
dy/du = d/du (sqrt(u))
= 1/(2sqrt(u))
Now, let's apply the chain rule:
dy/dx = (dy/du) * (du/dx)
= (1/(2sqrt(u))) * (sqrt(x) * sqrt(x-1) * (x-1)^(sqrt(x)-1))
Substituting back u = x^sqrt(x), we have:
dy/dx = (1/(2sqrt(x^sqrt(x)))) * (sqrt(x) * sqrt(x-1) * (x-1)^(sqrt(x)-1))
Simplifying this further would depend on your desired level of simplification. However, this equation will give you the derivative of Y with respect to x for the given function.
For the second problem:
(ii) Y = xarcsin(x) + sqrt(1-x)
To find dy/dx, we will again use the rules of differentiation.
Let's find the derivative of each term separately:
d/dx (xarcsin(x)) can be found using the product rule:
= x * d(arcsin(x))/dx + arcsin(x) * d(x)/dx
= x * d(arcsin(x))/dx + arcsin(x) * 1
= x * (1/sqrt(1 - x^2)) + arcsin(x)
d/dx (sqrt(1-x)) can be found using the power rule:
= (1/2) * d(1-x)^0.5/dx
= (1/2) * (1/2) * (1-x)^-0.5 * -1
= -1/2(1-x)^0.5
Now, let's find dy/dx by adding the derivatives of each term:
dy/dx = d/dx (xarcsin(x) + sqrt(1-x))
= x * (1/sqrt(1 - x^2)) + arcsin(x) - 1/2(1-x)^0.5
So, the derivative of Y with respect to x for the given function is:
dy/dx = x * (1/sqrt(1 - x^2)) + arcsin(x) - 1/2(1-x)^0.5