For this problem, consider the function
y=f(x)=x3
x2 – 1
To solve this problem, we need to find the derivative of the function y = f(x) = x^3(x^2 – 1).
Step 1: Expand the function using the distributive property:
y = x^3(x^2) - x^3(1)
= x^5 - x^3
Step 2: Differentiate each term with respect to x.
For the term x^5, we can apply the power rule. The power rule states that when differentiating a term of the form x^n, the derivative is given by n*x^(n-1).
Applying the power rule to x^5, we get:
d/dx(x^5) = 5x^(5-1)
= 5x^4
For the term x^3, we can also apply the power rule:
d/dx(x^3) = 3x^(3-1)
= 3x^2
Step 3: Combining the derivatives of each term:
The derivative of y = f(x) = x^3(x^2 – 1) is given by:
dy/dx = d/dx(x^5) - d/dx(x^3)
= 5x^4 - 3x^2
Therefore, the derivative of the function y = f(x) = x^3(x^2 – 1) is dy/dx = 5x^4 - 3x^2.