Log Chain

Log Chain

Chain Rule question.....u(x) = sin((log(x))) what is the derivative of u(x)?

Please explain the use of the chain rule I do not understand this rule at all but I know the rules for derivatives. Please explain each step. Thanks!

u(x) = sin((log(x))).

Say that v(x) = log(x). Then,

u(v) = sin(v)., and dv/dx = 1/x

Now, differentiate with respect to v.

du/dv = cos(v)

Then, du/dx = (du/dv)(dv/dx) = (cos(v))(1/x) = cos(v)/x

Now, substitute v = log(x), then you obtain,

du/dx = cos(log(x))/x

**Another way to understand it,... sort of. it's practically same
Keep in mind how when x is differentiated it becomes dx, u into du, v into dv, etc.

Then, when you have

u = sin(log(x)) or
u = sin(v)

differentiating both sides will result in

du = cos(v)dv ...................(i)

Now, since v = log(x), differentiating both sides results in

dv = (1/x)dx. ....................(ii)

Then, substituting (ii) into (i)

du = cos(v)dv
= cos(log(x))(1/x)dx.

Transfer, dx to the left, and you have

du/dx = cos(log(x))/x