MATH SOLVE

4 months ago

Q:
# find the derivative using quotient rule y = (x + 1)^2(x2 + 1)^-3 step by step pls

Accepted Solution

A:

let f(x)=g(x)/h(x)

The quotient rule states that:

f'(x)=[g'(x)h(x)-g(x)h'(x)]/[h(x)]²

From the expression given:

y = (x + 1)^2(x2 + 1)^-3

y=(x+1)²/(x²+1)³

where:

g(x)=(x+1)²

g'(x)=2(x+1)

h(x)=(x²+1)³

h'(x)=6x(x+1)²

hence to get the derivative of y we substitute in the formula:

y'=[2(x+1)(x²+1)³-6x(x+1)²(x+1)²]/[(x²+1)^6]

simplifying the above we get:

y[2(x+1)(x²+1)³-6x(x+1)^4]/[(x²+1)^6]

The quotient rule states that:

f'(x)=[g'(x)h(x)-g(x)h'(x)]/[h(x)]²

From the expression given:

y = (x + 1)^2(x2 + 1)^-3

y=(x+1)²/(x²+1)³

where:

g(x)=(x+1)²

g'(x)=2(x+1)

h(x)=(x²+1)³

h'(x)=6x(x+1)²

hence to get the derivative of y we substitute in the formula:

y'=[2(x+1)(x²+1)³-6x(x+1)²(x+1)²]/[(x²+1)^6]

simplifying the above we get:

y[2(x+1)(x²+1)³-6x(x+1)^4]/[(x²+1)^6]