Differentiate.
f(x)=5sqrt(x)sinx
Just use the good old chain rule and product rule:
f = 5√x sinx
f' = 5(1/(2√x) sinx + √x cosx)
= 5sinx/(2√x) + 5√x cosx
To differentiate the function f(x) = 5√(x)sin(x), we'll need to use the product rule and the chain rule.
The first step is to identify the different parts of the function. In this case, we have two parts - 5√(x) and sin(x). Let's label them as u(x) = 5√(x) and v(x) = sin(x).
Now, we can apply the product rule, which states that if we have two functions u(x) and v(x), their derivative is given by the following formula:
(d/dx)(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)
Let's calculate the derivatives of u(x) and v(x).
The derivative of u(x) = 5√(x) can be obtained using the chain rule. The chain rule states that if we have a composite function, f(g(x)), then the derivative is given by:
(d/dx)(f(g(x))) = f'(g(x))g'(x)
In our case, f(x) = 5√(x) and g(x) = x. The derivative of f(x) with respect to g(x) can be calculated using the power rule and the constant rule. The power rule states that if we have a function of the form f(x) = ax^n, then its derivative is given by:
(d/dx)(ax^n) = anx^(n-1)
Applying the power rule, we get:
f'(g(x)) = 5(1/2)√(x)^(1/2 - 1)
Simplifying further, we get:
f'(g(x)) = 5(1/2)√(x)^(-1/2)
Since the derivative of g(x) = x is simply 1, we can substitute these values into the product rule formula:
(d/dx)(u(x)v(x)) = (5(1/2)√(x)^(-1/2))sin(x) + (5√(x))(cos(x))
Simplifying further, we get:
(d/dx)(u(x)v(x)) = (5/2)(√(x)^(-1/2))sin(x) + (5√(x))(cos(x))
This is the derivative of f(x) = 5√(x)sin(x).