How to Find the Derivative of the Function f(x) = √x * cos(x) Using the Product Rule and Chain Rule

If f(x)=√xcosx, then f′(x)=

To find the derivative of the function f(x) = √x * cos(x) (denoted as f'(x)), we will use the product rule and the chain rule

To find the derivative of the function f(x) = √x * cos(x) (denoted as f'(x)), we will use the product rule and the chain rule.

The product rule states that for two functions u(x) and v(x), the derivative of their product is given by:
(uv)’ = u’v + uv’

In this case, let’s denote u(x) = √x and v(x) = cos(x).

Applying the product rule:

f'(x) = (√x)’ * cos(x) + √x * (cos(x))’

Now, let’s calculate the derivatives of u(x) and v(x) using the chain rule.

First, we differentiate u(x) = √x:
To do this, we can rewrite √x as x^(1/2).

So, u(x) = x^(1/2).

Applying the power rule:
u'(x) = (1/2) * x^(-1/2)

Next, let’s differentiate v(x) = cos(x):
Using the derivative of cosine, which is -sin(x):
v'(x) = -sin(x)

Now, substituting these derivatives into the equation for f'(x):

f'(x) = (1/2) * x^(-1/2) * cos(x) + √x * (-sin(x))

Simplifying this expression, we can write the final answer as:

f'(x) = (1/2) * cos(x) * x^(-1/2) – √x * sin(x)

So, the derivative of the function f(x) = √x * cos(x) is f'(x) = (1/2) * cos(x) * x^(-1/2) – √x * sin(x).

More Answers: