Does the derivative df/dg of the function f(g) exist and what does this mean?

[imy786]

Homework Statement

when is this statement true : the derivative df/dg of the function f(g) exists

what does this mean exactly?

Homework Equations

The Attempt at a Solution

does it mean-

1. df/dg represents the rate of change of f with respect to g at any given value
of g.

or

2.df/dg is the ratio of two small differences, df and dg, either of which may be positive or negative.

0r

3.df/dg is a function, called a derivative, which is always less than the value of
f(g) at any given value of g, that is, df/dg < f for all values of g.

i think the solution is 1.

because- definition of differentiation is:

Differentiation is a method to compute the rate at which a dependent output y changes with respect to the change in the independent input x. This rate of change is called the derivative of y with respect to x. In more precise language, the dependence of y upon x means that y is a function of x.

 

[imy786]

any hints?

im i correct or incorrect!

 

[Mark44]

Calculus problems should be posted in "Calculus & Beyond," not in the Precalculus section.

 

[imy786]

sorry

 

[Mark44]

To answer your question - it's 1. df/dg represents the derivative of f with respect to g.

For example, let f(x) = 2x + 3, and g(x) = x², and let h(x) = f(g(x)).

Then h'(x) (or dh/dx) = d/dx[f(g(x))] = f'(g(x)) * g'(x). The expression on the left here corresponds to what your problem calls df/dg.

Working in the example, we have h(x) = f(x²) = 2x² + 3, so h'(x) = 4x, using a direct approach.

Using the chain rule, we have h'(x) = f'(x²) * 2x = 2 * 2x = 4x.