Derivative of y=cos(xy)

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To find the derivative of this function, you'll need implicit differentiation.
derivative of y is just y'
derivative of cos(xy) requires chain rule AND product rule. It's -sin(xy)*(y+xy')

After taking this derivative, you'll need to multiply the -sin(xy) through the brackets (distribute) and then collect terms that contain y' on the one side of the equation. Then factor, and isolate y' with division.
  1. mroldridge
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Комментарии
Автор

Nice video! Just have a few observations:

a) It would be fun to write the function y as:
y = )...)




b) We can also use the multivariate chain rule to do essentially the same as what u did:

df(x, y) = (∂f/∂x) dx + (∂f/∂y) dy. In this example, with f(x, y) = cos(xy),

==> 1 = (∂f/∂x) (1/y') + (∂f/∂y) = [-ysin(xy)]/y' - xsin(xy)

==> 1/y' = [1+ xsin(xy)] / [-ysin(xy)], giving the same answer you got

adwz
Автор

Nice video ❤
I wonder, would this question be the same as asking:
Find the derivative of y for
y = cos(xcos(xcos(xcos(x…))))? Going infinitely

haidaralhassan
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Should note the domain of y is relative to the x given (arccos(y)/y)=x.

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