Question

Find the first derivative for each in the form

of dy/dx
First Problem (Reminder: simplify the numerator).
y= x²-1 / x³+1

Answer 1

The first derivative of the function y = (x² - 1) / (x³ + 1) is found using the quotient rule, resulting in a simplified expression after applying the rule and simplifying the numerator.

Step-by-step explanation:

To find the first derivative dy/dx of the given function y= x2 −1 / x3+1, you can use the quotient rule. The quotient rule states that if you have a function u/v , then the derivative is given by:

d/dx(u/v) = u'v-uv'/v2

Let's find the derivative for the given function:

y= x2 −1/x3 +1

Where u=x2 −1 and v=x3 +1.

Now, find the derivatives u′ and v′ :

u′ = 2x

v′ =3x2

Now apply the quotient rule:

dy/dx = (2x)(x 3+1)−(x2 −1)(3x2)/(x3 +1)2

Simplify the numerator:

dy/dx = 2x4+2x−(3x4 −3x2 )/(x3 +1)2

Combine like terms in the numerator:

dy/dx = −x4+3x2+2x/(x3 +1)2

So, the first derivative dy/dx for the given function is:

dy/dx = −x4+3x2+2x/(x3 +1) 2