derivatives?!?!?

Question

using derivatives: find all point on the graph of f(x)= 100/ x^5 where the tangent line is perpendicular to the line y=x

Answer 1

Take the first derivative,

f'(x) = -500/x^6

To be perpendicular to the line y = x which has a slope of 1, the tangent lines must have a slope of -1.

f'(x) = -1 or -500/x^6 = -1

x^6 = 500

x=+/- sqrt(500) = +/- 2.817269

Answer 2

the derivative of a function is defined as the rate at which the function is changing. For any given point on a curve, this is known as the slope of a line tangent to that point. The function y = x has a slope of dy = (1)x^(0)dx or dy/dx = 1
A function that is parallel to this function at any point will have the same slope. A function that is perpendicular at any point will have a slope that is the negative inverse of the y=x slope, ie. -1.

Take the function f(x) and compute its derivative

df(x)/dx = -500/x^6

Now set df(x)/dx = -1 manipulate the equation to find out what x is:

each answer will have an x and y coordinate. The 2 x coordinates will be plus and minus the 6th root of 500. the y coordinates will both be y = 100/(sixth root of x)^5. the y will be negative for the negative x value, and positive for the plus x value.


(-2.8173,-0.5634),(2.8173,0.5634)