Application of Derivatives

Question

Find the equation of the tangent and normal to the given curves at the points given:
$straight y space equals fraction numerator 5 straight x squared over denominator 1 plus straight x squared end fraction$

Solution

The equation of curve is $straight y space equals space fraction numerator 5 straight x squared over denominator 1 plus straight x squared end fraction$
$therefore space space space space dy over dx space equals space fraction numerator left parenthesis 1 plus straight x squared right parenthesis. space space 10 straight x minus space 5 straight x squared.2 straight x over denominator left parenthesis 1 plus straight x squared right parenthesis squared end fraction space equals space fraction numerator 10 straight x plus 10 straight x cubed minus 10 straight x cubed over denominator left parenthesis 1 plus straight x squared right parenthesis squared end fraction space equals space fraction numerator 10 straight x over denominator left parenthesis 1 plus straight x squared right parenthesis squared end fraction$
$At space left parenthesis 2 comma space 4 right parenthesis comma space space dy over dx space equals space fraction numerator 10 space cross times space 2 over denominator left parenthesis 1 plus 4 right parenthesis squared end fraction space equals space 20 over 25 space equals space 4 over 5 comma space space which space is space slope space of space tangent. therefore space space space space space space the space equation space of space tangent space at space left parenthesis 2 comma space 4 right parenthesis space is space space space space space space space straight y minus 4 space equals space 4 over 5 left parenthesis straight x minus 2 right parenthesis comma space space space or space space space 5 straight y minus 20 space equals space 4 straight x minus 8 space space space or space space 4 straight x minus 5 straight y plus 12 space equals space 0 Slope space of space normal space space equals space minus 5 over 4 therefore space space space space space space the space equation space of space normal space at space left parenthesis 2 comma space 4 right parenthesis space is space space space space space space space space space space straight y minus 4 space equals space minus 5 over 4 left parenthesis straight x minus 2 right parenthesis space comma space space space space or space space 4 straight y minus 16 space equals negative 5 straight x plus 10 or space space space space 5 straight x plus 4 straight y minus 26 space equals space 0$

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Application Of Derivatives

Find the equation of the tangent and normal to the given curves at the points given:
$straight y space equals fraction numerator 5 straight x squared over denominator 1 plus straight x squared end fraction$

Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

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