I am learning derivatives and I despise having to use the definition of the derivative. Please check my answer for this particular one! it is different from the one I get by differentiating.

The question: f(x) = 2+x / x^3
my answer: -6x^2-2x^2

Learn to use parentheses where needed. Is the numerator x or 2+x ?

sorry!

the numerator is 2+x and the denominator is x^3

To check your answer, let's first differentiate the function f(x):

Using the quotient rule, we differentiate the numerator and denominator separately and apply the formula:

f(x) = (2 + x) / x^3

f'(x) = [(x^3) * (d/dx)(2 + x) - (2 + x) * (d/dx)(x^3)] / (x^3)^2

Now, let's differentiate each term:

(d/dx)(2 + x) = 1

(d/dx)(x^3) = 3x^2

Plugging these results into the formula:

f'(x) = [(x^3) * 1 - (2 + x) * 3x^2] / x^6

Simplifying:

f'(x) = [x^3 - 3x^2(2 + x)] / x^6

Expanding the expression:

f'(x) = [x^3 - 6x^3 - 3x^3] / x^6

Combining like terms:

f'(x) = -8x^3 / x^6

Simplifying further:

f'(x) = -8 / x^3

So, the correct answer for the derivative of f(x) is f'(x) = -8 / x^3.

Comparing it to your answer, -6x^2 - 2x^2, it seems to be different. Please recheck your work and calculations.