Derivatives Equation

Question

Use the constant, power term, and sum of terms derivative rules to differentiate the following.

f(x) = (5x^3 + 3x^2 + 1) / (5x^2)

Answer 1

With any derivative problem. if you can simplify the function BEFORE starting to take the derivative, your life will be so much easier and less error-prone.

So for this one, I would divide each term of the numerator by the denominator, making a polynomial.

f(x) = [ 5x^3 / 5x^2 ] + [ 3x^2 / 5x^2 ] + [ 1 / 5x^2 ]

simplifying, this can be written as

f(x) = x + 3/5 + (1/5) x^(-2)

Now the power rule is all you need.

f '(x) = 1 + 0 + (1/5) (-2) x^(-3)

f ' (x) = 1 - [ 2 / (5x^3) ]

Answer 2

f = (5x^3+3x^2+1)/5x^2 = x+3/5+.2x^-2 ... f'= 1+0-.4x^-3