How do you prove the quotient rule?
By the definition of the derivative,
by taking the common denominator,
by switching the order of divisions,
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The quotient rule is a method used in calculus to find the derivative of a quotient of two functions. It states that if you have a function (f(x) = \frac{g(x)}{h(x)}), where both (g(x)) and (h(x)) are differentiable and (h(x) \neq 0), the derivative of (f(x)) with respect to (x) is:
[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{[h(x)]^2} ]
To prove the quotient rule, we start from the definition of the derivative and apply the limit process. Here’s a step-by-step proof:
- Definition of the derivative: The derivative of a function (f(x)) at a point (x) is defined as:
[ f'(x) = \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} ]
- Apply this definition to (f(x) = \frac{g(x)}{h(x)}):
[ f'(x) = \lim_{\Delta x \to 0} \frac{\frac{g(x + \Delta x)}{h(x + \Delta x)} - \frac{g(x)}{h(x)}}{\Delta x} ]
- Find a common denominator to combine the fractions:
[ f'(x) = \lim_{\Delta x \to 0} \frac{g(x + \Delta x)h(x) - g(x)h(x + \Delta x)}{\Delta x \cdot h(x + \Delta x) \cdot h(x)} ]
- Use the product rule on the numerator: Recall that the product rule states that the derivative of the product of two functions (u(x)v(x)) is (u'(x)v(x) + u(x)v'(x)). We apply a similar reasoning to handle the limit of a product as the product of limits, assuming (h(x) \neq 0) and (h(x + \Delta x) \neq 0):
[ f'(x) = \lim_{\Delta x \to 0} \frac{g(x + \Delta x)h(x) - g(x)h(x + \Delta x)}{\Delta x} \cdot \frac{1}{h(x + \Delta x) \cdot h(x)} ]
- Apply the limit definition of the derivative to (g(x)) and (h(x)):
By recognizing the expression (\frac{g(x + \Delta x) - g(x)}{\Delta x}) as the derivative of (g(x)) and similarly for (h(x)), we can rewrite the expression to include (g'(x)) and (h'(x)), along with the remaining terms involving (g(x)) and (h(x)):
[ f'(x) = \frac{g'(x)h(x) - g(x)h'(x)}{h(x)^2} ]
This step incorporates the derivatives (g'(x)) and (h'(x)), recognizing that the terms involving (\Delta x) in the numerator cancel out and approach (g'(x)) and (h'(x)) respectively as (\Delta x) approaches 0, while the denominator simplifies to (h(x)^2).
This completes the proof of the quotient rule, demonstrating how to derive the formula for the derivative of a quotient of two functions using the definition of the derivative and limit properties.
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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.
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.
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