Question

Y=√a+√a+√a+x^2
Find derivative

Answer 1

Answer:

1/4 . 1/[a + (a+x)½ ]½ . 1/(a+x)½

Step-by-step explanation:

Substitution Method:

y = [a + (a+x)½]½

Put u = a + (a+x)½ . Therefore,

y = u½ Differentiating both sides with respect to x,

dy/dx = d/du(u½) . du/dx = 1/2 . u(½-¹) . d/dx [a + (a+x)½ ]

=(1/2) . u(-½).[da/dx + d/dx (a+x)½ ]

Substituting for u,

dy/dx = 1/2 . 1/[a + (a+x)½ ]½ . [0 + (1/2).(a+x)-½] . d/dx(a+x) (since a is a constant)

= 1/2 . 1/[a + (a+x)½ ]½ . 1/2.(a+x)½ .( da/dx + dx/dx)

=1/2 . 1/[a + (a+x)½ ]½ . 1/2.(a+x)½ . (0+1)

=1/4[a + (a+x)½ ]½ . [(a+x)½]

Direct Method Without Substitution:

y = [a + (a+x)½]½ Differentiate directly and write (function of a function)

dy/dx = d/dx[a + (a+x)½]½ . d/dx[a + (a+x)½]

= (1/2) .[a + (a+x)½]-½ . [da/dx + d/dx{(a+x)½}]

dy/dx = 1/2[a + (a+x)½]½ . [da/dx + d/dx[(a+x)½] . d/dx [(a+x)] Since a is a constant, da/dx=0 and

= 1/{2[a + (a+x)½]½} . 1/2 . (a+x)-½ . d/dx [(a+x)]

= 1/4 . 1/[a + (a+x)½]½ . 1/(a+x)½ . (da/dx + dx/dx)

= 1/4 . 1/[a + (a+x)½]½ . 1/(a+x)½ . 1

= 1/4[a + (a+x)½]½ [1/(a+x)½]