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

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

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

That is,

By signing up, you agree to our Terms of Service and Privacy Policy

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

By signing up, you agree to our Terms of Service and Privacy Policy

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}]

By signing up, you agree to our Terms of Service and Privacy Policy

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7