Find the equation of the line tangent to the graph of y = sin^-1 (x/5) at x = 5/2. Show your work.

Step-by-step explanation:

We want to find the equation of the line tangent to the graph of:

So, we will find the derivative of our equation first. Applying the chain rule, we acquire that:

Simplify:

We can factor out the denominator within the square root:

Simplify:

So, we can find the slope of the tangent line at x = 5/2. By substitution:

Evaluate:

We will also need the point at x = 5/2. Using our original equation, we acquire that:

So, the point is (5/2, π/6).

Finally, by using the point-slope form, we can write:

Distribute:

Isolate. Hence, our equation is:

x =±sqrt(1/3)

step-by-step explanation:

x+4/6x=1/x

multiply each side by 6x to get rid of the fractions

6x*(x+4/6x)=1/x*6x

distribute

6x^2 +4 = 6

subtract 4 from each side

6x^2 +4-4 = 6-4

6x^2 =2

divide by 6

6x^2/6 = 2/6

x^2 = 1/3

take the square root of each side

sqrt(x^2) = ±sqrt(1/3)

x =±sqrt(1/3)