Question

How do you find the tangent line of `f(x) = 3x/sinx^2` at x=5?

Answer 1

`y = -754.2x + 3806`

Explanation:

Use the quotient rule when your function can be thought of as one function divided by another.

if `f(x) = (g(x))/(h(x))`
then `f'(x) = (g'(x)h(x)-g(x)h'(x))/(h^2(x))`

so if `f(x) = (3x)/sinx^2`
then `f'(x) = (3sinx^2 - 3x(2xcosx^2))/sin^2x^2`

so `f'(x) = (3sinx^2 - 6x^2cosx^2)/sin^2x^2`

Now I'm gonna assume we're working in degrees because `x=5` seems more like a value for degrees than radians, but I could be wrong. If I am the steps still apply it's just that my numbers will be off.

The derivative will give us the gradient, `m`, of the tangent, `y=mx+c`, at any given point `x`.

`m = (3sin25-150cos25)/(sin25)^2 = -134.7/0.1786`

`m = -754.2`

If you work with radians you should get around `-8520` as your gradient or something.

Now we have the equation

`y = -754.2x + c`

for which we still need the value for `c`, which we can obtain by substituting a point `(x,y)` which is on the tangent line.

We are already given a point `x=5`, so we should work out the corresponding `y` value:

`y = (3x)/(sinx^2) =15/sin25 = 35.49`

`:.`

`(x,y)=(5.000, 35.49)`

which we can substitute in to give us

`35.49 = -754.2 xx 5.000 + c`

`35.49 + 3771 = 3806 = c`

so

`y = -754.2x + 3806`

Obviously this will be a very different equation using radians, but the general principles are the same. I've also used all values to 4 significant figures.