Find an equation of the tangent line to the curve at the given point.
y= (9x)/(x+1)^2 P = (0,0)
dy/dx= 9/(x+1)^2+9x(-2)/(x+1)^3=m
solve for m at x=0,
dy/dx= 9
y= mx+b
at (o,0), b=0
so
y=9x is the equation for the line.
check my work.
To find the equation of the tangent line to the curve at the point (0,0), we need to find the slope of the tangent line first.
The slope of the tangent line can be determined by taking the derivative of the function y = (9x)/(x+1)^2 with respect to x.
Let's find the derivative:
Using the quotient rule, the derivative of y with respect to x is:
y' = [(9(x+1)^2)(1) - (9x)(2(x+1)(1))]/(x+1)^4
= (9(x+1) - 18x)/(x+1)^3
= (9x + 9 - 18x)/(x+1)^3
= (9 - 9x)/(x+1)^3
Now that we have the derivative, we can find the slope of the tangent line at the point (0,0) by substituting x = 0 into y':
m = (9 - 9(0))/(0+1)^3
= (9 - 0)/1
= 9
Therefore, the slope of the tangent line at the point (0,0) is 9.
Now, we can use the point-slope form of a line to find the equation of the tangent line:
y - y1 = m(x - x1)
Plugging in the values, we have:
y - 0 = 9(x - 0)
Simplifying,
y = 9x
Therefore, the equation of the tangent line to the curve y = (9x)/(x+1)^2 at the point (0,0) is y = 9x.
To find the equation of the tangent line to the curve at the given point, you can follow these steps:
1. Find the derivative of the given function with respect to x. This will give you the slope of the tangent line at any point on the curve.
So let's find the derivative of y = (9x)/(x+1)^2.
Using the quotient rule, the derivative is given by:
dy/dx = [(9)(x+1)^2 - (9x)(2)(x+1)] / (x+1)^4
Simplifying further, we have:
dy/dx = (9(x+1) - 18x(x+1)) / (x+1)^3
= (9x + 9 - 18x^2 -18x) / (x+1)^3
= (-18x^2 - 9x + 9) / (x+1)^3
2. Plug in the x-coordinate of the given point (P) into the derivative to find the slope of the tangent line at that specific point.
Since P is (0,0), we substitute x=0 into the derivative:
dy/dx = (-18(0)^2 - 9(0) + 9) / (0+1)^3
= 9 / 1
= 9
So the slope of the tangent line at point P is 9.
3. Use the point-slope form of the equation of a line to find the equation of the tangent line.
The point-slope form is given by: y - y1 = m(x - x1), where (x1, y1) are the coordinates of the point on the line and m is the slope.
Plugging in the values from point P, we have:
y - 0 = 9(x - 0)
y = 9x
Thus, the equation of the tangent line to the curve y = (9x)/(x+1)^2 at the point (0,0) is y = 9x.