Consider the function f(x) = (2x^3 + x^2 - 5x) / (x + 2).
a) Use the Product Rule to find the derivative of f(x).
b) Use the Quotient Rule to find the derivative of f(x).
a) To find the derivative of f(x) using the Product Rule, we can set f(x) as the product of two functions: u(x) = 2x^3 + x^2 - 5x and v(x) = 1/(x + 2).
The Product Rule states that for two differentiable functions, u(x) and v(x), the derivative of their product f(x) = u(x) * v(x) is given by:
f'(x) = u'(x) * v(x) + u(x) * v'(x)
Let's find the derivatives of u(x) and v(x):
u'(x) = d/dx (2x^3 + x^2 - 5x) = 6x^2 + 2x - 5
v'(x) = d/dx (1/(x + 2)) = -1/(x + 2)^2
Now, we can use the Product Rule to find the derivative of f(x):
f'(x) = (6x^2 + 2x - 5) * (1/(x + 2)) + (2x^3 + x^2 - 5x) * (-1/(x + 2)^2)
Simplifying this expression further, we can write:
f'(x) = (6x^2 + 2x - 5)/(x + 2) - (2x^3 + x^2 - 5x)/(x + 2)^2
So, the derivative of f(x) using the Product Rule is given by (6x^2 + 2x - 5)/(x + 2) - (2x^3 + x^2 - 5x)/(x + 2)^2.
b) To find the derivative of f(x) using the Quotient Rule, we can set f(x) as the quotient of two functions: u(x) = 2x^3 + x^2 - 5x and v(x) = (x + 2).
The Quotient Rule states that for two differentiable functions, u(x) and v(x), the derivative of their quotient f(x) = u(x) / v(x) is given by:
f'(x) = (u'(x) * v(x) - u(x) * v'(x)) / v(x)^2
Let's find the derivatives of u(x) and v(x) again:
u'(x) = 6x^2 + 2x - 5
v'(x) = 1
Now, we can use the Quotient Rule to find the derivative of f(x):
f'(x) = ((6x^2 + 2x - 5) * (x + 2) - (2x^3 + x^2 - 5x) * 1) / (x + 2)^2
Expanding and simplifying this expression further, we get:
f'(x) = (6x^3 + 12x^2 + 2x^2 + 4x - 5x - 10 - 2x^3 - x^2 + 5x) / (x + 2)^2
f'(x) = (4x^3 + 14x^2 - 11) / (x + 2)^2
Therefore, the derivative of f(x) using the Quotient Rule is given by (4x^3 + 14x^2 - 11) / (x + 2)^2.
Note: It is always a good practice to simplify the final derivative expression as much as possible.