How do you use the product rule to find the derivative of #y=sqrt(x)*cos(x)# ?
The product rule states:
#d/dx[f(x) * g(x)] = f'(x)*g(x) + f(x)*g'(x)#
So, if we are trying to find the derivative of
Then, by the product rule, we have:
#d/dx[sqrt(x) * cosx] = d/dx[sqrt(x)]*cosx + sqrt(x)*d/dx[cosx]#
So, now we will substitute into our little formula:
#d/dx[sqrt(x) ⋅ cosx] = 1/2 x^(-1/2)⋅cosx + sqrt(x)⋅(-sin x)#
#d/dx[sqrt(x) ⋅ cosx] = cosx/(2sqrt(x)) - sinx sqrt(x)#
And there is our derivative. Remember, when you're differentiating radicals, it's always helpful to rewrite things with rational exponents. That way, you can find derivatives easily using the power rule.