Quotient Rule:
Use the limit definition of the derivative to prove that the quotient rule
google is your friend. You can find what you need here:
http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeProofs.aspx
Why did the mathematician turn to comedy? Because he wanted to prove that the quotient rule was actually a joke! But if you insist on a more formal explanation, let me break it down for you.
To prove the quotient rule using the limit definition of the derivative, we start by considering the function f(x) = g(x) / h(x). We want to find the derivative of this function, which is given by f'(x).
Using the limit definition of the derivative, we have:
f'(x) = lim(h->0) [f(x + h) - f(x)] / h
Now let's simplify it using the structure of our function:
f'(x) = lim(h->0) [(g(x + h) / h(x + h)) - (g(x) / h(x))] / h
Now, let's do some fancy algebra:
f'(x) = lim(h->0) [(g(x + h) * h(x) - g(x) * h(x + h)) / (h(x + h) * h)]
Now, let's break it up into two separate limits:
f'(x) = lim(h->0) [g(x + h) * h(x) - g(x) * h(x + h)] / lim(h->0) h(x + h) * h
Now, we can use the limit properties to evaluate the two limits separately:
f'(x) = [g'(x) * h(x) - g(x) * h'(x)] / h(x)^2
And there you have it, my friend! The quotient rule proven using the limit definition of the derivative. Remember, math jokes are the best way to calculate a good laugh!
To prove the quotient rule using the limit definition of the derivative, we need to start with the definition of the derivative. The derivative of a function f(x) at a point x=a is defined as:
f'(a) = lim(x→a) (f(x) - f(a))/(x - a)
Let's consider two differentiable functions u(x) and v(x), and we want to find the derivative of their quotient, f(x) = u(x)/v(x). We can express this as:
f(x) = u(x) * 1/v(x)
To find the derivative of f(x), we will apply the limit definition of the derivative.
Step 1: Start by finding the difference quotient of f(x) using the definition of the derivative:
f'(x) = lim(h→0) (f(x + h) - f(x))/h
Step 2: Substitute in the expression for f(x) using u(x) and v(x):
f'(x) = lim(h→0) ((u(x + h)/v(x + h)) - (u(x)/v(x)))/h
Step 3: Simplify the expression by multiplying both the numerator and the denominator by the conjugate of the second term:
f'(x) = lim(h→0) ((u(x + h)v(x) - u(x)v(x + h))/(v(x + h)v(x)))/h
Step 4: Expand the numerator:
f'(x) = lim(h→0) (u(x + h)v(x) - u(x)v(x + h))/(h * v(x + h)v(x))
Step 5: Distribute the limit operator over each term in the numerator:
f'(x) = lim(h→0) u(x + h)v(x)/(h * v(x + h)v(x)) - lim(h→0) u(x)v(x + h)/(h * v(x + h)v(x))
Step 6: Apply the limit to each term separately:
f'(x) = lim(h→0) u(x + h)v(x)/(h * v(x + h)v(x)) - lim(h→0) u(x)v(x)/(h * v(x + h)v(x))
Step 7: Now, use the limit definition of the derivative for each term:
f'(x) = u'(x)v(x)/v(x)^2 - u(x)v'(x)/v(x)^2
Step 8: Combine the terms and place them over a common denominator:
f'(x) = (u'(x)v(x) - u(x)v'(x))/v(x)^2
And there we have the quotient rule:
f'(x) = (u'(x)v(x) - u(x)v'(x))/v(x)^2
Therefore, we have successfully proven the quotient rule using the limit definition of the derivative.
To prove the quotient rule using the limit definition of the derivative, we need to express the derivative of a quotient of functions as a limit.
Let's say we have two differentiable functions f(x) and g(x), where g(x) is not equal to zero for all x in the domain. We want to find the derivative of their quotient f(x) / g(x), denoted as (f/g)'.
Using the limit definition of the derivative, we have:
(f/g)' = lim(h->0) [ (f(x + h) / g(x + h)) - (f(x) / g(x)) ] / h
Now, we need to simplify this expression using the algebraic properties of limits:
(f/g)' = lim(h->0) [ (f(x + h) * g(x) - f(x) * g(x + h)) / (g(x + h) * g(x)) ] / h
Next, we can multiply the numerator and denominator by g(x) to eliminate the fraction inside the limit:
(f/g)' = lim(h->0) [ (f(x + h) * g(x) - f(x) * g(x + h)) / (g(x + h) * g(x)) ] * (1/g(x)) * (1/g(x)) / h
Simplifying further, we get:
(f/g)' = lim(h->0) [ (f(x + h) * g(x) - f(x) * g(x + h)) / (g(x + h) * g(x)) ] * (1/g(x))^2 / h
Now, we can distribute the limit to both terms in the numerator:
(f/g)' = lim(h->0) [ (f(x + h) * g(x) - f(x) * g(x + h)) / (g(x + h) * g(x)) ] * lim(h->0) (1 / (g(x))^2) / h
Taking the limit as h approaches 0, we can cancel out the h term:
(f/g)' = lim(h->0) [ (f(x + h) * g(x) - f(x) * g(x + h)) / (g(x + h) * g(x)) ] * lim(h->0) (1 / (g(x))^2)
Simplifying the expression further, we can rewrite it as:
(f/g)' = [ f'(x) * g(x) - f(x) * g'(x) ] / (g(x))^2
This is the quotient rule for differentiation.
So, by using the limit definition of the derivative and simplifying step by step, we have derived the quotient rule (f/g)' = [ f'(x) * g(x) - f(x) * g'(x) ] / (g(x))^2.