using product rule let y=uv

Question

a)  (x²+1)³(x³+1)²    b)   (x²-1)√(1+x)      c)   (√1/x-x²)(x²+x)⁷

Answer 1

Assuming this is calculus, these problems involve both the product rule and chain rule.

Product rule: If y = uv, then y' = u'v + uv'

Chain rule d(f(u))/dx = f'(u) * u'

 

I will work through b, and hopefully you can use that as a guide for a and c.

 

y = (x²-1)√(1+x)

y = (x²-1)(1+x)^1/2

 

Splitting this into factors for the product rule:

u = (x²-1)

u' = 2x

 

The second factor requires chain rule, although the derivative of the inside is 1 in this case. In both a and c, the inner derivatives are more complicated.

v = (1+x)^1/2

v' = (1/2)(1+x)^-1/2 * (1)

v' = 1/(2√(1+x))

 

Put together:

y' = u'v + uv'

y' = (x²-1) * 1/(2√(1+x)) + (2x) * (1+x)^1/2

y' = (x²-1)/(2√(1+x)) + (2x)√(1+x)