How do you find an equation of the tangent line to the curve at the given point: #y=(2x)/(x+1)^2# at point (0,0)?

Answer 1
Find an equation of the tangent line to the curve at the given point: #y=(2x)/(x+1)^2# at point (0,0)
Find #y'# using the quotient rule, the power rule, and the chain rule:
#y'=((2)(x+1)^2-(2x)(2(x+1)*(1)))/(x+1)^4#
# =(2(x+1)^2-4x(x+1))/(x+1)^4#
We could do some more algebra, but it is clear at this point that when #x=0#, we have
#y'=2/1=2#
So the tangent line is through #(0,0)# and has slope #2#:
#y=2x#
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Answer 2

To find the equation of the tangent line to the curve at the point (0,0), we need to find the derivative of the function y=(2x)/(x+1)^2 and evaluate it at x=0. The derivative of the function is given by the quotient rule, which states that the derivative of (f(x)/g(x)) is (f'(x)g(x) - f(x)g'(x))/(g(x))^2. Applying the quotient rule to the given function, we get:

y' = [(2(x+1)^2 - 2x(2(x+1))]/((x+1)^2)^2

Simplifying this expression, we have:

y' = [2(x+1)^2 - 4x(x+1)]/[(x+1)^2]^2

Evaluating this expression at x=0, we get:

y'(0) = [2(0+1)^2 - 4(0)(0+1)]/[(0+1)^2]^2

Simplifying further, we have:

y'(0) = [2(1) - 4(0)]/[1]^2

y'(0) = 2/1

y'(0) = 2

Therefore, the slope of the tangent line at the point (0,0) is 2.

Using the point-slope form of a linear equation, y - y1 = m(x - x1), where (x1, y1) is the given point and m is the slope, we can substitute the values x1=0, y1=0, and m=2 into the equation to find the equation of the tangent line:

y - 0 = 2(x - 0)

Simplifying this equation, we have:

y = 2x

Therefore, the equation of the tangent line to the curve y=(2x)/(x+1)^2 at the point (0,0) is y = 2x.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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