The main purpose of this question is to find the differential of each given function.
A function is a fundamental mathematical concept that describes a relationship between a set of inputs and a set of possible outputs, with each input corresponding to one output. The input is an independent variable and the output is referred to as a dependent variable.
Differential calculus and integral calculus are the fundamental classifications of calculus. Differential calculus deals with infinitely small changes in some varying quantity. Let $y=f(x)$ be a function with a dependent variable $y$ and an independent variable $x$. Let $dy$ and $dx$ be the differentials. The differential forms the main part of the change in a function $y = f(x)$ as the independent variable changes. The relation between $dx$ and $dy$ is given by $dy=f'(x)dx$.
More generally, differential calculus is used to investigate the instantaneous rate of change, for instance, velocity, to estimate the value of a small variation in a quantity, and to determine whether a function in a graph is increasing or decreasing.
(a) The given function is:
Here, $y$ is dependent and $t$ is an independent variable.
Taking differential of both sides using the chain rule as:
(b) The given function is:
Here, $y$ is dependent and $v$ is an independent variable.
Taking differential of both sides using the quotient rule as:
Find the differential of the following functions:
Using the power rule on first term and the chain rule on second term as:
Using power rule on all the terms as:
Rewrite the function as:
Now use the power rule on all the terms as:
Rewrite the given function as:
Now use power rule on all the terms as:
(e) $y=\ln(\sin (2x))$
Using the chain rule as:
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