- #1
michonamona
- 122
- 0
Homework Statement
Is the composition of two differentiable functions always differentiable?
E.x.
h(x) = sin(x)
k(x) = 1/x for x not equal 0
Does this automatically mean h(k(x)) is differentiable?
Thank you,
M
Sure. You even know a formula for the derivative, right?
I'll just comment about one of my little pet peeves. h(k(x)) is a number, not a function. The function you have in mind is written as LaTeX Code: h\\circ k or LaTeX Code: x\\mapsto h(k(x)) . (Note the special "mapsto" arrow).
The composition of two differentiable functions, denoted as (f ∘ g)(x), is a new function that results from applying one function (g) to the output of another function (f). In other words, the output of f becomes the input of g.
The composition of two differentiable functions can be calculated using the chain rule, which states that the derivative of the composition of two functions is equal to the product of the derivatives of each individual function.
When two functions are differentiable, it means that they have a well-defined derivative at every point in their domain. This allows us to calculate the derivative of the composition of two functions using the chain rule.
Yes, it is possible for the composition of two differentiable functions to be non-differentiable. This can occur when the two functions have a discontinuity or a point where the derivative is undefined in their composition.
The composition of two differentiable functions is commonly used in physics, engineering, and economics to model real-world phenomena. For example, in physics, the composition of position and velocity functions can be used to calculate the acceleration of an object at a given point in time.