Show that the relative rate of change of a quotient f / g is the difference between the relative

Show that the relative rate of change of a quotient f / g is...

Key Concepts

Relative Rate of Change

The relative rate of change of a function is the ratio of its derivative to the function itself. It measures how quickly the function's value is changing per unit of its current value, providing a normalized rate of change that is especially useful in comparing the growth or decay of different quantities.

Logarithmic Differentiation

Logarithmic differentiation involves taking the natural logarithm of a function before differentiating, which is particularly useful when the function is a product, quotient, or power. This method simplifies the differentiation process by converting multiplicative relationships into additive ones; for example, the derivative of ln(f/g) becomes the difference between the derivatives of ln f and ln g.

Differentiation of Quotient Functions (Quotient Rule)

The differentiation of quotient functions is guided by the quotient rule, which provides a systematic way to compute the derivative of a ratio of two functions. When combined with the concept of relative rate of change, the quotient rule helps in showing that the relative rate of change of a quotient is equal to the difference of the relative rates of change of the numerator and denominator.

Transcript

0:00 Hi there.

00:02 In this problem, we're asked to show that the relative rate of change, i'll just write rroc, the relative rate of change of a quotient, so f divided by g, and these are functions, so really it's f of x over g of x.

00:21 We're supposed to show that it's the difference of the two relative rates.

00:24 In other words, we're supposed to show that it's the relative rate of change of f minus the relative rate of change of g.

00:36 One thing we can really do and that to try to figure out the left -hand side what is a relative rate of change of f over g so we'll have to start by recalling our definition of relative rate of change okay the relative rate of change of some function is defined as its derivative it over the original function so in our case the relative rate of change of f over g i guess is just defined as let's take its derivative on top, so f over g will need its derivative all over f over g.

01:21 That's just the definition of a relative rate of change.

01:24 We take the thing as derivative and divided by the thing itself.

01:29 Okay.

01:32 Now, on top, notice we're taking the derivative of a quotient, so we're definitely going to need the quotient rule.

01:39 And let's remember what the quotient rule is from this chapter.

01:44 The quotient rule says that the derivative will be g.

01:47 F prime minus f g prime all over g squared so that is just using the quotient rule and then the bottom is still f over g okay so at this point it's all algebra so remember dividing by a fraction is the same as multiplying by its reciprocal so why don't we for now i'm going to leave the top alone but instead of dividing by all this, let's multiply by g over f.

02:31 Okay...

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