Each of the following expressions of a variable \(x\)
can be thought of as the formula for a real-valued function
that’s continuous on some open subset of the real numbers.
As exercise, for each of these functions,
write down an explicit formula for its derivative.
Use software capable of symbolic differentiation to check your formulas.
Each of the following equations involves an independent variable \(x\)
and a variable \(y\) dependent on \(x.\)
As exercise, for each of these equations,
take the derivative with respect to \(x\) implicitly.
Then if possible, express the derivative of \(y\)
explicitly as a function of \(x\).
Use software capable of symbolic differentiation to check your formulas.
\(\displaystyle \tan\Bigl(\cos\bigl(\cos(xy)\bigr)\Bigr) = y \)
Now suppose each of the previous equations
involved an independent variable \(t\) (not explicitly present)
and variables \(x\) and \(y\) each dependent on \(t.\)
As exercise, for each of those equations equations,
take the derivative with respect to \(t\) implicitly.
Use software capable of symbolic differentiation to check your formulas.