Question

How do you use the product rule to find the derivative of `y=sqrt(x)*cos(x)` ?

Answer 1

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 `y = sqrt(x) * cosx`, then we will let `f(x) = sqrt(x)` and `g(x) = cosx`.

Then, by the product rule, we have:

`d/dx[sqrt(x) * cosx] = d/dx[sqrt(x)]*cosx + sqrt(x)*d/dx[cosx]`

`sqrt(x)` is the same thing as `x^(1/2)`. Therefore, by the power rule, the derivative of `x^(1/2)` is `1/2 x^(-1/2)`. And, we know from basic trig derivative rules that `d/dx[cosx] = -sinx`.

So, now we will substitute into our little formula:

`d/dx[sqrt(x) ⋅ cosx] = 1/2 x^(-1/2)⋅cosx + sqrt(x)⋅(-sin x)`

Recalling that `x^(-1/2)` is the same thing as `1/sqrt(x)`, we will simplify this equation:

`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.