# Derivative with respect to Time

*Derivatives and Bank Collapse**But just as much as it is easy to find the derivative of a given quantity, so it is difficult to find the integral of a given derivative. Moreover, sometimes we cannot say with certainty whether the integral of a given quantity can be found or not.*(Johann Bernoulli)*Among all of the mathematical disciplines the theory of differential equations is the most important… It furnishes the explanation of all those elementary manifestations of nature which involve time.*(Sophus Lie)

We are now ready to give a formal definition of the *derivate* of a function depending on time .

**Definition:** A function defined on an interval , is said to be *differentiable* in with *derivative* if for some positive constant

- .

A differentiable function is locally close to a linear function in in the sense that up to a quadratic term in .

A differentiable function is Lipschitz continuous, since it is locally close to a linear function and a linear function is Lipschitz continuous.

We note the following connection between Lipschitz continuity and the size of the derivative.

**Theorem:** If is differentiable with for , then is Lipschitz continuous with Lipschitz constant .

**Proof:** This result should be intuitively obvious: It is like saying that if your velocity is never bigger than 1 km/hour, then it is impossible to travel a distance longer than 1 km in an hour. Right? If you hesitate, consider the follwoing formal proof:

Given , we are supposed to show that

- .

To this end let us note that it is sufficient to prove that for any given ,

- .

Now choose first and then such that and for some natural number , assuming . We then have splitting the interval into subintervals of length : , ,…., and using the definition of on each subinterval:

- ,

that is, using that ,

- .

By the triangle inequality we now have since ,

- ,

which we wanted to show.

# Example: Kink

The function is Lipschitz continous (in particular at ), but is not differentiable at because it is not close to a linear function for close to (since it has a kink).

# Read More

- \hyperref[derivative]{Derivative.}
- \hyperref[diffrules]{Rules of Differentiation.

# To Remember:

- A Lipshitz continuous function is locally close to a constant function.
- A differentiable function is locally close to a linear function

Next: Derivative with respect to Space Previous: Lipschitz Continuity

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