Find f^'(0) and f^'(1). f(x)=(a x+b)/(c x+d) ; a, b, c, d ...

Key Concepts

Quotient Rule

The quotient rule is a fundamental technique in calculus for differentiating functions that are ratios of differentiable functions. It states that the derivative of a function expressed as a quotient f(x) = g(x)/h(x) is given by f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x))^2. This rule is essential when dealing with rational functions like the one in the problem.

Linear Fractional Transformation

A linear fractional transformation is a specific type of rational function of the form f(x) = (ax + b)/(cx + d). These functions are also known as Möbius transformations and are widely studied for their unique mapping properties. Understanding their structure simplifies the process of differentiation, especially since both the numerator and denominator are linear functions.

Evaluation of Derivatives at Specific Points

After computing the derivative of a function, it is often necessary to evaluate it at particular points to understand the instantaneous rate of change at those points. This involves substituting the desired values into the differentiated function. It is a common technique used in calculus for analyzing the behavior of functions at specific inputs.

Transcript

00:01 So we're looking at this function, a x plus b divided by cx plus d.

00:05 We're going to find this function's derivative first.

00:07 So f prime of x.

00:09 It's going to be a quotient rule here.

00:11 So it's going to be the derivative of the numerator, ax plus b, multiplied by the denominator, which is cx plus d, and then minus the numerator, multiplied by the derivative of the denominator, and divided by our denominator squared.

00:31 So really we just need to find the derivative of a x plus b and cx plus d or the derivative of the numerator and the denominator so for a more general i guess if we're talking about a general quotient rule you're always going to need to find the derivative of the numerator and the denominator and everything else is just going to be either the numerator or the denominator so let's go ahead and find the derivative of the numerator that's going to be equal to the derivative of a x plus the derivative of b which is just equal to a, and then the derivative of cx plus d is going to be equal to the derivative of cx plus the derivative of d, which is equal to c.

01:14 So we have these two values here.

01:17 I'm just going to make this smaller and put it up here.

01:21 And so we can say this derivative is equal to a times cx plus d minus c times a x plus b divided by cx plus d squared and now that we have our functions derivative we're going to find the value of this derivative at zero and then at one so for f prime of zero this is going to be equal to a times c times zero would be zero plus d so a times d minus c times a times zero to be zero plus b...

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