How do you differentiate #[(2x^3) - (4x^2) + 3] / x^2 #?

Question

Cameron Alexander · Accepted Answer

#d/dx ((2x^3 - 4x^2 + 3)/x^2)= 2 -6 /x^3#

While it would be possible to use the quotient rule on this rational function, as it can easily be achieved in this particular case, it will be more convenient (and simpler, less prone to error, and quicker in an exam) to divide each term in the numerator by the term in the denominator and then use the sum rule.

Thus

#(2x^3 - 4x^2 + 3)/x^2#

#= 2x - 4 + 3x^(-2)#

which may be differentiated term by term using the rules for simple polynomial differentiation.

That is,

#d/dx (2x - 4 + 3x^(-2)) #

#= 2 -6 x^(-3)#

#= 2 -6 /x^3#

Emma Aldridge · Answer

undefined To differentiate the expression [(2x^3) - (4x^2) + 3] / x^2, you would use the quotient rule of differentiation. The quotient rule states that if you have a function f(x) divided by a function g(x), the derivative of f(x) divided by g(x) is equal to the derivative of f(x) times g(x) minus f(x) times the derivative of g(x), all divided by g(x) squared.

Applying this rule to the given expression, you would first find the derivative of the numerator and denominator separately. Then, apply the quotient rule formula to obtain the derivative of the entire expression.

The derivative of the numerator (2x^3 - 4x^2 + 3) with respect to x is:
d(2x^3 - 4x^2 + 3)/dx = 6x^2 - 8x

The derivative of the denominator (x^2) with respect to x is:
d(x^2)/dx = 2x

Now, applying the quotient rule:
[(6x^2 - 8x)(x^2) - (2x^3 - 4x^2 + 3)(2x)] / (x^2)^2

Simplifying further:
(6x^4 - 8x^3 - 4x^4 + 8x^3 - 6x) / x^4
(2x^4 - 6x) / x^4

Which simplifies to:
2 - 6/x

Emma Aldridge · Answer

undefined To differentiate the expression $(2x^3 - 4x^2 + 3) / x^2$, you can use the quotient rule of differentiation, which states that if you have a function in the form $f(x) = \frac{g(x)}{h(x)}$, then its derivative is given by $$f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{(h(x))^2}.$$

Applying the quotient rule to the given expression:

$$f(x) = \frac{2x^3 - 4x^2 + 3}{x^2}$$

$$f'(x) = \frac{(2(3x^2) - 4(2x))(x^2) - (2x^3 - 4x^2 + 3)(2x)}{(x^2)^2}$$

$$f'(x) = \frac{(6x^2 - 8x)(x^2) - (2x^3 - 4x^2 + 3)(2x)}{x^4}$$

$$f'(x) = \frac{6x^4 - 8x^3 - 2x^4 + 8x^3 - 8x^2 + 12x}{x^4}$$

$$f'(x) = \frac{4x^4 - 8x^2 + 12x}{x^4}$$

$$f'(x) = 4 - \frac{8}{x^2} + \frac{12}{x^3}$$