Derivations in calculus
Question 1.
In this case we are going to use first principle of differentiation to differentiate our function;
We now use logarithms to further compute our problem to re-write right hand side first.
Further using the law of
Now to derive we need to let tend to zero. Now substituting we need to let
Proceeding with
And so
But
Now it’s clear that
After clear illustration of how we have derived the we can post some questions that fellow classmate can use to compute.
- Show using the first principles, that if then .
Chain rule.
This rule is known as the chain rule since we use it to take derivatives of composites of functions by chaining together their derivatives. The chain rule can be thought of as taking the derivatives of the external function (applied to the inner function) and duplicating it times the derivatives of the inner function.
Now to differentiate we re-write this expression in its alternative form using logarithms;
Now differentiating in respect to x both sides we have
Now using chain rule
and so
Rearranging,
But
After clear illustration of how we have derived the we can post some questions that fellow classmate can use to compute.
- Show using the first principles, that if then .
Question 3.
Logistic Function
A logistic function or logistic curve is a typical S-formed curve with a condition where = the estimation of the sigmoid’s midpoint, = the curve’s most extreme worth, = the logistic development rate, or steepness of the curve.
The logistic function is the reverse of the common logit function thus can be utilized to change over the logarithm of chances into a likelihood. In scientific documentation, the logistic function is some of the time composed as a master in a similar structure as logit.
Post on Application.
A biologist starts with 100 bacteria in an observation. After 3 days he discovers that the population has grown to 350.
Determine an equation for this bacteria population.