The quotient rule is a formula for finding the derivative of the ratio of two differentiable functions. It states that if $u(x)$ and $v(x)$ are functions, then $(\frac{u}{v})' = \frac{u'v - uv'}{v^2}$.
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The numerator of the quotient rule formula is formed by subtracting the product of the derivative of the denominator function and the numerator function from the product of the derivative of the numerator function and the denominator function.
The denominator in the quotient rule formula is always the square of the denominator function.
The quotient rule can be remembered by 'low d high minus high d low over low squared'.
It’s essential to correctly apply both differentiation rules and algebraic manipulation when using the quotient rule.
Mistakes often occur when students forget to square the denominator or misapply negative signs.
Review Questions
What is the quotient rule formula for differentiating $\frac{f(x)}{g(x)}$?
When applying the quotient rule, what common mistakes should you avoid?
Given $u(x) = x^2 + 3x$ and $v(x) = x - 1$, find $(\frac{u}{v})'$ using the quotient rule.
Related terms
Product Rule: A differentiation rule used to find derivatives of products of two functions: $(uv)' = u'v + uv'$.
Chain Rule: A fundamental differentiation rule stating that if a variable y depends on a variable u, which itself depends on x, then y also depends on x, and dy/dx = (dy/du) * (du/dx).
Power Rule: A basic differentiation rule that states if $f(x) = x^n$, then $f'(x) = nx^{n-1}$.