Differentiate the functions with respect to x.
\(sec(tan(\sqrt x))\)
Differentiate the functions with respect to x.
\(sec(tan(\sqrt x))\)
Solution and Explanation
Let f(x) = sec (tan(\(\sqrt x\)), u(x) = \(\sqrt x\), and v(t) = tan t and w(s) = sec s
Then, (wovou)(x) = w[v(u(x))] = w[v(√x)] = w(tan(\(\sqrt x\))) = sec (tan(\(\sqrt x\))) = f(x)
Thus, f is a composite function of three functions u, v, and w.
Put s = v(t) = tan and t = u(x) = \(\sqrt x\)
Then, we obtain
\(\frac {dw}{ds}\) = \(\frac {d}{ds}\) (sec s) = sec s.tan s = sec (tan t) . tan (tan t) [s = tan t]
= sec (tan\(\sqrt x\)) .tan (tan\(\sqrt x\))
\(\frac {ds}{dt}\) = \(\frac {d}{dt}\)(tan t) = sec2t = sec2\(\sqrt x\)
\(\frac {dt}{dx}\) = \(\frac {dt}{dx}\)(\(\sqrt x\)) = \(\frac {dt}{dx}\)(\(x^{\frac 12}\)) = \(\frac 12\) . \(x^{\frac {1}{2}-1}\) = \(\frac {1}{2\sqrt x}\)
Therefore by chain rule, \(\frac {dt}{dx}\) = \(\frac {dw}{ds}\) . \(\frac {ds}{dt}\) . \(\frac {dt}{dx}\)
= sec (tan\(\sqrt x\)) . tan (tan\(\sqrt x\)) . sec2\(\sqrt x\) . \(\frac {1}{2\sqrt x}\)
= \(\frac {1}{2\sqrt x}\) sec2\(\sqrt x\) . sec (tan\(\sqrt x\)) . tan (tan\(\sqrt x\))
= \(\frac {sec\ (tan\sqrt x) . tan\ (tan\sqrt x) . sec^2\sqrt x}{2\sqrt x}\)
Alternate method:
\(\frac {d}{dx}\) [sec(tan(\(\sqrt x\)))] = sec (tan\(\sqrt x\)) . tan (tan\(\sqrt x\)) . \(\frac {d}{dx}\) (tan\(\sqrt x\))
= sec (tan\(\sqrt x\)) . tan (tan\(\sqrt x\)) . sec2\(\sqrt x\) . \(\frac {d}{dx}\)\((\sqrt x)\)
= sec (tan\(\sqrt x\)) . tan (tan\(\sqrt x\)) . sec2\((\sqrt x)\) . \(\frac {1}{2\sqrt x}\)
= \(\frac {sec\ (tan\sqrt x) . tan\ (tan\sqrt x) . sec^2\sqrt x}{2\sqrt x}\)
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Notes on Continuity and differentiability
Concepts Used:
Derivatives
Derivatives are defined as a function's changing rate of change with relation to an independent variable. When there is a changing quantity and the rate of change is not constant, the derivative is utilised. The derivative is used to calculate the sensitivity of one variable (the dependent variable) to another one (independent variable). Derivatives relate to the instant rate of change of one quantity with relation to another. It is beneficial to explore the nature of a quantity on a moment-to-moment basis.
Few formulae for calculating derivatives of some basic functions are as follows:
