How do you differentiate f(x)= ln(sin(x^2)/x) ?

1 Answer
Dec 30, 2015

To differentiate the ln, we'll need quotient rule.
To differentiate sin(x^2), we'll need chain rule as well.
To differentiate sin(x^2)/x, we'll need quotient rule.

Explanation:

  • Chain rule: (dy)/(dx)=(dy)/(du)(du)/(dx)
  • Quotient rule: be y=f(x)/g(x), then y'=(f'g-fg')/g^2

We can rename u=sin(x^2)/x so that f(x)=ln(u), which is differentiable.

Also, we can rename v=x^2 so we can differentiate sin(v) applying chain rule, as well.

(dy)/(dx)=1/u*(2xcos(x^2)*x-sin(x^2)*1)/x^2

(dy)/(dx)=(2x^2cos(x^2)-sin(x^2))/(xsin(x^2))

Recalling trigonometric identities: cota=cosa/sina

Let's split the result a bit.

(dy)/(dx)=(2x^cancel(2)color(green)(cos(x^2)))/(cancel(x)color(green)(sin(x^2)))-cancel((sin(x^2)))/(xcancel(sin(x^2)))

(dy)/(dx)=2xcot(x^2)-1/x