let f(x)= 6sin(x)/(2sin(x)+6cos(x))

The equation of the tangent line to y= f(x) at a= pi/6 can be written in tthe form y=mx+b where m=? b=?
I found the f':
(9*(cos^2(x)+sin^2(x)))/(3cos(x)+sin(x))^2
but i don't know how to fill it into the form. Can anyone help me?
Thanks you

now, make x=PI/6, and calculate f'

Then, y=f'(PI/6)*x + b

your f ' (x) is correct but not complete

f ' (x) = (9*(cos^2(x)+sin^2(x)))/(3cos(x)+sin(x))^2
remember that sin^2 x + cos^2 x = 1

f '(x) = 9/(3cosx + sinx)^2

so when x = π/6 ,
f'(π/6) = 9/(3√3/2 + 1/2)^2
= appr .93769

this is the slope of the tangent when x = π/6
when x = π/6
f(π/6) = 3/(1+3√3) = appr .4842

equation of tangent:
y - .4842 = .93769(x - .5236)

check my arithmetic

To find the equation of the tangent line to the curve y = f(x) at a given point (a, f(a)), you need to find the slope of the tangent line (m) and the y-intercept (b).

1. Start by finding the derivative of f(x), which you have correctly calculated as f'(x) = (9*(cos^2(x)+sin^2(x)))/(3cos(x)+sin(x))^2.

2. Substitute the value of the point (a, f(a)) into the derivative to find the slope. In this case, a = pi/6, so substitute pi/6 into the derivative: f'(pi/6) = (9*(cos^2(pi/6)+sin^2(pi/6)))/(3cos(pi/6)+sin(pi/6))^2.

3. Calculate the value of f'(pi/6) to find the slope of the tangent line.

4. After finding the slope, substitute the values of the point (a, f(a)) into the equation y = mx + b to find the y-intercept, b.

Let's go through the steps:

Step 1: Calculating f'(x):
The derivative you have provided is correct: f'(x) = (9*(cos^2(x)+sin^2(x)))/(3cos(x)+sin(x))^2.

Step 2: Substituting a = pi/6 into f'(x):
f'(pi/6) = (9*(cos^2(pi/6)+sin^2(pi/6)))/(3cos(pi/6)+sin(pi/6))^2.

Step 3: Evaluating f'(pi/6):
To find the slope of the tangent line, you need to evaluate the derivative at x = pi/6.
f'(pi/6) = (9*(cos^2(pi/6) + sin^2(pi/6)))/(3cos(pi/6) + sin(pi/6))^2 = (9*(1/2 + 3/4))/(3(√3/2) + 1/2)^2 = (9*(5/4))/(3(√3/2) + 1/2)^2.

Step 4: Finding the y-intercept (b):
Now that you have the slope of the tangent line (m), which is f'(pi/6), you can substitute the values of (a, f(a)) = (pi/6, f(pi/6)) into the equation y = mx + b to solve for b.

So, the equation of the tangent line can be written in the form:
y = f'(pi/6) * x + b,

where m = f'(pi/6) and b is the y-intercept you find by substituting the values of (a, f(a)).