Find the equation of the tangent line to the curve y=(5x)/(1+x^2) at the point (5,0.96154).The equation if this tangent line can be written in the form y= mx +b. Find m and b

sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to remember what the formula of tangent line to a curve, at a point, is:

`y - f(x|_(x=a)) = f'(x|_(x=a))(x - a)`

You need to differentiate the function with respect to x, using quotient rule, such that:

`f'(x) = ((5x)'*(1+x^2)-(5x)*(1+x^2)')/((1+x^2)^2)`

`f'(x) = (5 + 5x^2 - 10x^2)/((1+x^2)^2)`

`f'(x) = (5-5x^2)/((1+x^2)^2)`

You need to substitute 5 for x in `f'(x)`  such that:

`f'(5) = (5 - 5*25)/((1+25)^2)`

`f'(5) = -120/676 `

`f'(5) = -30/169`

You need to substitute 5 for x, 0.96154 for f(5) and -0.17751 for f'(5) in equation `y - f(x|_(x=a)) = f'(x|_(x=a))(x - a) ` such that:

`y - f(5) = f'(5)(x-5)`

`y - 0.96154 =-0.17751(x-5)`

You need to open brackets such that:

`y - 0.96154 = -0.17751x + 0.88757`

You need to isolate y to the left side such that:

`y =-0.17751x + 0.88757 + 0.96154`

`y =-0.17751x + 1.84911`

Hence, evaluating the equation of tangent line to the curve, at the point `(5,0.96154)`  yields `y = -0.17751x + 1.84911.`