Differentiate!!!!?

Question

f(x) = cos(ln x)

Answer 1

y = cos u
u = ln x
du/dx = 1 / x
y = cos u
dy/du = - sin u
dy/dx = (dy/du) (du/dx)
dy/dx = - [sin (ln x) ] (1/x)

Answer 2

The derivative of cosx is -sinx
The derivative of lnx is 1/x

When a function is embedded in another, you write the embedded function again, then multiply it by the derivative of the interior function. This is the definition of the chain rule.

So..
f(x) = g(h(x))
f'(x) = g'(h(x))*h'(x)
In this case, h(x) is lnx and g(x) is cos x


f(x) = cos(ln x)
f'(x) = -sin(ln x)*1/x

Good luck.

Answer 3

f'(x) = -sin(ln(x)) * (1/x)
= -sin(ln(x))/x

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Long version:
y = cos(ln(x))
Let u = ln(x)
du/dx = 1/x

y = cos(u)
dy/du = -sin(u) = -sin(ln(x))

dy/dx = dy/du * du/dx
= -sin(ln(x)) * 1/x
= -sin(ln(x))/x