differentiate y=(x)^(1/2)e^(-x).i need the calculation work. [ans: -x^(1/2)e^(-x) + 1/2(x^-1/2)e^(-x)]
Use the product rule.
d/dx (u*v) = u(dv/dx) + v(du/dx)
u = x^(1/2)
du/dx = (1/2)*x(-1/2)
v = e^-x
dv/dx = -e^-x
You do the rest
To differentiate the given function y = (x)^(1/2) e^(-x), we will need to use the product rule and the chain rule.
The product rule states that if we have two functions, u(x) and v(x), then the derivative of their product is given by:
(d/dx)(u*v) = u * (d/dx)(v) + v * (d/dx)(u)
For our function, let's consider u(x) = (x)^(1/2) and v(x) = e^(-x).
First, let's find the derivative of u(x) using the chain rule. The chain rule states that if we have a composition of functions, F(g(x)), then the derivative is given by:
(d/dx)(F(g(x))) = (dF/dg) * (dg/dx)
For u(x) = (x)^(1/2), we can rewrite it as u(x) = x^(1/2) = (x)^(0.5).
Now, let's use the chain rule to find the derivative of u(x):
(d/dx)(u(x)) = (d/dx)(x^(1/2)) = (1/2) * (x^(1/2 - 1)) * (d/dx)(x) = (1/2) * x^(-1/2) * 1 = 1/2x^(1/2)
Next, let's find the derivative of v(x) = e^(-x). The derivative of e^x is simply e^x, and since we have a negative sign in front of x, the derivative of e^(-x) will be -e^(-x).
Now, using the product rule, we can find the derivative of y = (x)^(1/2) e^(-x):
(d/dx)(y) = u * (d/dx)(v) + v * (d/dx)(u)
= (x)^(1/2) * (-e^(-x)) + e^(-x) * (1/2x^(1/2))
Simplifying the expression further, we get:
(d/dx)(y) = -x^(1/2) * e^(-x) + 1/2(x^(-1/2)) * e^(-x)
Hence, the derivative of y = (x)^(1/2) e^(-x) is -x^(1/2) * e^(-x) + 1/2(x^(-1/2)) * e^(-x).
Note: It's always a good practice to double-check the obtained derivative by plugging in some values of x and comparing the result with the given answer.