Question

How do you differentiate `f(x)=1/ln(sinx)` using the quotient rule?

Answer 1

Step 1: Use the chain rule to differentiate the denominator:

Let `y = ln(u)` and `u = sinx`.

Then `y' = 1/u` and `u' = cosx`

`dy/dx = dy/(du) xx (du)/dx`

`dy/dx = 1/u xx cosx`

`dy/dx = cosx/sinx`

`dy/dx = cotx`

Now, we have the derivative of the denominator, which will be useful when we use the quotient rule.

Step 2: Use the quotient rule:

Let `f(x) = g(x)/(h(x))`

Then `g(x) = 1` and `h(x) = ln(sinx)`. The derivative of `g(x)`, since it's a constant is `0` and the derivative of `h(x)` is `cotx`, as shown in step 1.

The quotient rule states that `f'(x) = (g'(x) xx h(x) - g(x) xx h'(x))/(h(x))^2`.

`f'(x) = (0 xx ln(sinx) - 1 xx cotx)/(ln(sinx))^2`

`f'(x) = -(cotx)/(ln(sinx))^2`

Hopefully this helps!