What is the equation of the tangent line of #f(x) =(4x^3) / ((3-5x)^5) # at # x = 2#?

Answer 1

Tangent line equation is
#color(red)(y=(464x)/117649-1152/117649)#

Also

#color(red)(464x-117649y=1152)#

Determine the point of tangency #(x_1, y_1)# then the slope #m#
From the given equation #f(x) = (4x^3)/(3-5x)^5# #y_1=f(2)=(4(2)^3)/(3-5(2))^5=32/(-7)^5=-32/16807#
#color(red)((x_1, y_1)=(2, -32/16807))#
Obtain the slope #m#
#f(x) = (4x^3)/(3-5x)^5#
#f' (x)=((3-5x)^5d/dx(4x^3)-4x^3*d/dx(3-5x)^5)/((3-5x)^5)^2#
#f' (x)=((3-5x)^5*12x^2-4x^3*5(3-5x)^4*(-5))/((3-5x)^5)^2#
Use #x=2# to find the slope #m=f' (2)#
#f' (x)=((3-5x)^5*12x^2-4x^3*5(3-5x)^4*(-5))/((3-5x)^5)^2#
#f' (2)=((3-5(2))^5*12(2)^2-4(2)^3*5(3-5(2))^4*(-5))/((3-5(2))^5)^2#
#f' (2)=((-7)^5*48+32*25(-7)^4)/(-7)^10#
#f' (2)=((-7)*48+32*25)/(-7)^6#
#color(red)(f' (2)=(464)/117649)##color(red)" the slope"#

The Point-Slope Form's Tangent line

#(y-y_1)=m(x-x_1)#
#(y-(-32/16807))=(464/117649)(x-2)#
#y+32/16807=(464/117649)(x-2)#
#y=(464/117649)(x-2)-32/16807#
#color(red)(y=(464x)/117649-1152/117649)#

Also

#color(red)(464x-117649y=1152)#
Kindly see the graph, the tangent #464x-117649y=1152# is almost coincident with the x_axis. Take note the slope #m=0.003944~=0# at the point #(2, -0.0019)# graph{(y - (4x^3)/(3-5x)^5)(464x-117649y-1152)=0[-3,3,-1.5,1.5]}

May God bless you all. I hope this explanation helps.

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Answer 2

To find the equation of the tangent line of f(x) at x = 2, we need to find the slope of the tangent line and a point on the line.

First, we find the derivative of f(x) using the quotient rule:

f'(x) = [(3-5x)^5 * 12x^2 - 4x^3 * 5(3-5x)^4 * (-5)] / (3-5x)^10

Next, we substitute x = 2 into f'(x) to find the slope of the tangent line:

f'(2) = [(3-5(2))^5 * 12(2)^2 - 4(2)^3 * 5(3-5(2))^4 * (-5)] / (3-5(2))^10

Finally, we can simplify the expression to find the slope of the tangent line at x = 2.

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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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