Question

g(x)= sqrt(1-4x^2)

Can someone help me find the first and second derivatives please!

Answer 1

write it as

g(x) = (1 - 4x^2)^(1/2)
g ' (x) = (1/2)(1 - 4x^2)^(-1/2) (-8x)
= -4x(1 - 4x^2)^(-1/2)

now by the product rule

g '' (x)
= (-4x)(-1/2)(1-4x^2)^(-3/2) (-8x)- 4(1 - 4x^2)^(-1/2)
= (-16x^2)(1-4x^2)^(-3/2) - 4(1 - 4x^2)^(-1/2)
= -4(1 - 4x^2)^(-3/2) [ 4x^2 - (1-4x^2) ]
= -4(1-4x^2)^(-3/2) (4x^2 + x - 2)
= -4( 1-4x^2)^(-3/2)
or
= -4/(1 - 4x^2)^(3/2)