How to find the derivative of a function using the quotient rule?

The Quotient Rule for Derivatives

Now that you're familiar with basic differentiation rules, let's learn how to find the derivative of a function using the quotient rule. The quotient rule is used to differentiate a function in the form of a quotient, i.e., uv\frac{u}{v}, where uu and vv are functions of xx.

The Quotient Rule:

The quotient rule states that if uu and vv are functions of xx, then the derivative of uv\frac{u}{v} is given by:

ddx(uv)=vdudx−udvdxv2\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}

In other words, to find the derivative of a quotient, you multiply the denominator by the derivative of the numerator and subtract the product of the numerator and the derivative of the denominator, all divided by the square of the denominator.

Example:

Let's find the derivative of f(x)=x2x3+1f(x) = \frac{x^2}{x^3 + 1} using the quotient rule.

  1. Identify uu and vv:

    • u=x2u = x^2
    • v=x3+1v = x^3 + 1

  2. Find the derivatives of uu and vv:

    • dudx=2x\frac{du}{dx} = 2x
    • dvdx=3x2\frac{dv}{dx} = 3x^2

  3. Apply the quotient rule: ddx(x2x3+1)=(x3+1)(2x)−(x2)(3x2)(x3+1)2\frac{d}{dx}\left(\frac{x^2}{x^3 + 1}\right) = \frac{(x^3 + 1)(2x) - (x^2)(3x^2)}{(x^3 + 1)^2}

  4. Simplify the expression: ddx(x2x3+1)=2x4+2x−3x4(x3+1)2=−x4+2x(x3+1)2\frac{d}{dx}\left(\frac{x^2}{x^3 + 1}\right) = \frac{2x^4 + 2x - 3x^4}{(x^3 + 1)^2} = \frac{-x^4 + 2x}{(x^3 + 1)^2}