Find the derivative of the function.

y= xcosx - sinx

What's the derivative?

I get only one trig function. I will be happy to critique your work or thinking. Use the chain rule on the first term.

Our class hasn't covered Chain Rule. Only product and quotient rule. Apparently the work should go...

y= -xsinx + cosx - cosx

I get where the -xsinx and -cosx came from. But the +cosx?

Nevermind. I figured it out. Used the product rule on the xcosx, then found the derivative of the -sinx.

Good job on using the product rule to find the derivative of the term xcosx! The product rule states that if you have two functions, u(x) and v(x), then the derivative of their product is given by:

(d/dx)(u(x)v(x)) = u'(x)v(x) + u(x)v'(x)

In this case, u(x) = x and v(x) = cosx.

So, applying the product rule, we have:

(d/dx)(xcosx) = (d/dx)(x)cosx + x(d/dx)(cosx)

The derivative of x with respect to x is simply 1, so the first term becomes 1*cosx = cosx.

The derivative of cosx with respect to x is -sinx, so the second term becomes x*(-sinx) = -xsinx.

Therefore, the derivative of xcosx is cosx - xsinx.

Now, let's find the derivative of -sinx. The derivative of sinx with respect to x is simply cosx.

Therefore, the derivative of -sinx is -cosx.

Putting it all together, the derivative of the function y = xcosx - sinx is:

y' = (cosx - xsinx) - cosx

Simplifying the expression, we get:

y' = -xsinx

So, the derivative of the given function is -xsinx.