How do you differentiate f(x)=sqrtxsinx?

1 Answer
Jun 14, 2018

d/dx(sqrt(x)sin(x))=sin(x)/(2sqrt(x))+sqrt(x)cos(x)

Explanation:

To take the derivative of this, we need two rules. The first rule is the chain rule, which states:

d/dx(f(x)g(x))=f'(x)g(x)+f(x)g'(x)

The second rule we need is the power rule, which states:

d/dx(x^n)=nx^(n-1)

Now, we can start taking the derivative. To simplify this, I'm going to rewrite sqrt(x) as x^(1/2), which is perfectly valid.

To start, I'm going to take the derivative of each section.

d/dx(x^(1/2))=1/2x^(-1/2)

d/dx(x^(1/2))=1/(2sqrtx)

d/dx(sin(x))=cos(x)

Now that we have them, we can put them together:

d/dx(sqrt(x)sin(x))=sin(x)/(2sqrt(x))+sqrt(x)cos(x)

And there you have your answer.