0
The quotient rule tells says that:
If: f(x) = g(x) / h(x)
Then: f'(x) = (g'(x) · h(x) - g(x) · h'(x)) / h(x)2
That derivative gives you the slope of the original curve's tangent at any given x-coordinate.
For example, if:
f(x) = axn / bxm
then:
f'(x) = (anxn-1 · bxm - axn · bmxm - 1) / b2x2m
This of course simplifies considerably, due to the simplicity of the original function:
= (abnxm+n-1 - abmxm+n-1) / b2x2m
= (n - m)abxm+n-1 / b2x2m
= (n - m)axn-m-1 / b
= (n - m)(a/b)xn-m-1
Which makes perfect sense if you rearrange our initial function and take it's derivative in a simpler manner:
f(x) = axn / bxm
f(x) = (a/b)xn-m
f'(x) = (n-m)(a/b)xn-m-1
Still curious? Ask our experts.
Chat with our AI personalities
Professor
Taiga
LaoAdd your answer:
Earn +20 pts
What is a slope line?
is it a line that is slanted
How do you find x assume that segments that appear tangent are tangent?
A tangent refers to the way in which a curve is measured. The amount of deviation from the segment line is measures, then a formula applied to find the tangent.
How do you locate rate of change on a graph?
Find the slope of the tangent to the graph at the point of interest.
How do you find a tangent line?
Say you are given a function and an x value.(1) First find the y coordinate that corresponds to that x value by plugging x into the function and simplifying to find y = some #. Now you have a point (x, y) that is not only on the function, but also on the tangent line.(2) Take the derivative of the function.(3) If the derivative still has xs in it, plug in the x value you were given and simplify. This should give you an actual number--the slope of the tangent line.(4) From steps 1 and 3, you now have a point on the tangent line and the slope of the tangent line. Use these two things to write the equation for the tangent line in y=mx+b form (m is the sope, plug in the point you found, solve for b, then rewrite the equation replacing m and b but leaving in x and y).
Does a trigonometry tangent relate to a circle's tangent?
A circle's tangent is exactly the same as a triangle's tangent. If you look at a circle, you can make the radius the hypotenuse. Then make a vertical line from the point, and a horizontal line from the center. If you look, you have a triangle made inside the circle. This is why angles can be measured in radians, a unit that is derived from the circumference of a circle.-------------------------------------------------------------------------------------------By doing a little calculus, we find that the slope of the equation of a circle-the slope of the tangent line-is given by the tangent of an angle.AnswerEverything written above is correct, but doesn't have anything to do with tangents (in the circle sense of the word). Suppose you're given an angle theta. Draw a circle together with two radii, one horizontal and the other at an angle theta from the first one. (So far, this is the same as above.) Now draw the tangent to the circle at X, the point where the non-horizontal radius meets the circumference. Let Y be the point where this tangent meets the horizontal line through the centre. Then, assuming the radius is 1, tan(theta) is the distance XY, which is the length of part of the tangent.
