Question

63-78 Find the derivative of the function. Simplify where possible.

65. \(y = {\bf{ta}}{{\bf{n}}^{ - {\bf{1}}}}\sqrt {x - {\bf{1}}} \)

Short answer

The derivative of the function \(f\left( x \right)\) is \(\frac{1}{{2x\sqrt {x - 1} }}\).

Step-by-step solution

Write the differentiation formula for \(f\left( x \right)\)

Thedifferentiation of \({\tan ^{ - 1}}x\) is:

\(\frac{{\rm{d}}}{{{\rm{d}}x}}\left( {{{\tan }^{ - 1}}x} \right) = \frac{1}{{1 + {x^2}}}\)

Find the derivative of the \(f\left( x \right)\)

The differentiation of \(\frac{{{\rm{d}}y}}{{{\rm{d}}x}}\) is:

\(\begin{aligned}\frac{{{\rm{d}}y}}{{{\rm{d}}x}} &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {{{\tan }^{ - 1}}\sqrt {x - 1} } \right)\\ &= \frac{1}{{1 + {{\left( {\sqrt {x - 1} } \right)}^2}}} \times \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {\sqrt {x - 1} } \right)\\ &= \frac{1}{{1 + x - 1}} \times \frac{1}{{2\sqrt {x - 1} }}\\ &= \frac{1}{{2x\sqrt {x - 1} }}\end{aligned}\)

Thus, the derivative of the function y is \(\frac{1}{{2x\sqrt {x - 1} }}\).