Find the derivative of f(x)=(2x)^(1/2)ln(15x)
To find the derivative of f(x) = (2x)^(1/2)ln(15x), we can use the product rule and the chain rule.
The product rule states that if we have two functions u(x) and v(x), the derivative of their product is given by (u(x)v(x))' = u'(x)v(x) + u(x)v'(x).
Let's identify u(x) and v(x) in the given function:
u(x) = (2x)^(1/2)
v(x) = ln(15x)
To find the derivative, we need to find the derivatives of u(x) and v(x) individually.
Using the power rule, the derivative of u(x) = (2x)^(1/2) is given by:
u'(x) = 1/2 (2x)^(-1/2) (2) = (1/2)(2)(2x)^(-1/2) = (1/x)^(1/2) = (1/x)^(1/2) / 1 = 1/(2x)^(1/2)
Using the chain rule, the derivative of v(x) = ln(15x) is given by:
v'(x) = (1/15x)(15) = 1/x
Now, we can apply the product rule:
f'(x) = u'(x)v(x) + u(x)v'(x)
= (1/(2x)^(1/2)) * ln(15x) + (2x)^(1/2) * (1/x)
Simplifying further:
f'(x) = ln(15x) / (2x)^(1/2) + ( 2x)^(1/2) / x
Therefore, the derivative of f(x) = (2x)^(1/2)ln(15x) is f'(x) = ln(15x) / (2x)^(1/2) + ( 2x)^(1/2) / x.