Differentiate the following
y=root(e^x)/x
what kind of root? square root ?
is it √(e^x/x) or (√e^x) / x ?
y=(√e^x)/x
y=(√e^x)/x
y = (e^(x/2) )/x
dy/dx = (x(1/2)e^(x/2) - e^(x/2) )/x^2
simplify if needed.
To differentiate the function y = √(e^x)/x, we can use the quotient rule of differentiation. The quotient rule states that if we have a function of the form f(x)/g(x), where f(x) and g(x) are differentiable functions, then the derivative of this function can be found using the formula:
(d/dx)[f(x)/g(x)] = (g(x)*(d/dx)[f(x)] - f(x)*(d/dx)[g(x)]) / (g(x))^2
Now let's apply the quotient rule to differentiate y = √(e^x)/x:
Step 1: Identify the functions f(x) and g(x):
f(x) = √(e^x)
g(x) = x
Step 2: Find the derivatives (d/dx)[f(x)] and (d/dx)[g(x)]:
(d/dx)[f(x)] = (1/2)*(e^x)^(-1/2)*(d/dx)[e^x]
(d/dx)[g(x)] = 1
To find (d/dx)[f(x)], we need to use the chain rule. The chain rule states that if we have a composition of functions such as f(g(x)), then the derivative is given by (d/dx)[f(g(x))] = (d/dg)[f(g)] * (d/dx)[g(x)].
Applying the chain rule, we have:
(d/dx)[f(x)] = (1/2)*(e^x)^(-1/2)*(d/dx)[e^x] = (1/2)*(e^x)^(-1/2)*e^x = (1/2)*(e^x)^(1/2) = (1/2)*√(e^x)
Substituting these values into the quotient rule formula:
(d/dx)[√(e^x)/x] = (x*(1/2)*√(e^x) - √(e^x)*1) / x^2
= (x√(e^x))/(2x) - √(e^x)/x^2
= (√(e^x))/(2x) - √(e^x)/x^2
Therefore, the derivative of y = √(e^x)/x is (√(e^x))/(2x) - √(e^x)/x^2.