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

1 Answer
Jul 13, 2016

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!