Explanation d 2 y d x 2 is the second derivative of y with respect to x Given: x = s − 1 + s , y = 4 − s − 2 , is parametric form of the curve y = m x where s is the parameter. Formula used: d 2 y d x 2 = x ' ( s ) y ' ( s ) − y ' ( s ) x ' ' ( s ) ( x ' ( s ) ) 3 --------------------- (i) Where x ' ( s ) = d x ( s ) d s ; x " ( s ) = d 2 x ( s ) d s 2 y ' ( s ) = d y ( s ) d s ; y ' ' ( s ) = d 2 y ( s ) d s 2 d d s ( s n ) = n s n − 1 --------------------------- (ii) d d s ( c o n s tant ) = 0 -------------------- (iii) d d s ( a f ( s ) ) = a d d s f ( s ) ------------------- (iv), a is a constant. d d s ( a f ( s ) + b g ( s ) ) = a d d s f ( s ) + b d d s g ( s ) ----------- (v), a , b are constant. d d s ( s ) = 1 x ' ( s ) = d x ( s ) d s = d d s ( s − 1 + s ) = d ( s − 1 ) d s + d ( s ) d s , [using (v)] = ( − 1 ) s − 1 − 1 + 1 , [using (ii)] x ' ( s ) = − s − 2 + 1 x ' ' ( s ) = d 2 x ( s ) d s 2 = d d s [ d x ( s ) d s ] = d d s [ x ' ( s ) ] = d d s [ − s − 2 + 1 ] = − d d s ( s − 2 ) + d d s ( 1 ) , [using (v)] = − ( − 2 ) s − 2 − 1 + 0 = − 2 s − 3 , [using (ii)] x ' ' ( s ) =

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$\frac{d^{2} y}{d x^{2}}$

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d 2 y d x 2

Solution Summary: The author explains that d 2 y s x 2 is not defined. It is the second derivative of a curve with respect to m.

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$\frac{d^{2} y}{d x^{2}}$