how do you differentiate f(x) = 3(ln(x)) * (tan x)/(sin x + x ^ 3 + 8)

Question

how do you differentiate the following

f(x) = 3(ln(x)) * (tan x)/(sin x + x ^ 3 + 8) 

Answer 1

We would need to use the quotient rule, then the product rule for the top:

d/dx f(x) / g(x) = f'(x)g(x) - f(x) g'(x) / (g(x))² (quotient rule)

d/dx (a(x) b(x)) = a'(x)b(x) + b'(x) a(x)

In this case: f(x) = 3 ln x tan x and g(x) = sin x + x³ + 8 for the quotient rule:

We use the product rule for d/dx 3 ln x tan x, with a(x) = 3 ln x and b(x) = tan x

d/dx 3 ln x tan x = 3/x tan x + 3 sec²x ln x

This means if we use the quotient rule, the equation becomes:

(3/x tan x + 3 sec²x ln x) (sin x + x³ + 8) - 3 ln x tan x (cos x + 3x²)/ (sin(x) + x³ + 8)²

It could be more simplified, but I think you can handle that.