Question

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

66. \(y = {\bf{ta}}{{\bf{n}}^{ - {\bf{1}}}}\left( {{x^{\bf{2}}}} \right)\)

Short answer

The derivative of the function \(f\left( x \right)\) is \(\frac{{2x}}{{1 + {x^4}}}\).

Step-by-step solution

Write the differentiation formula for y

The differentiation 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 y

The differentiation of y with respect to xis:

\(\begin{aligned}\frac{{{\rm{d}}y}}{{{\rm{d}}x}} &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {{{\tan }^{ - 1}}\left( {{x^2}} \right)} \right)\\ &= \frac{1}{{1 + {{\left( {{x^2}} \right)}^2}}} \times \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {{x^2}} \right)\\ &= \frac{1}{{1 + {x^4}}} \times \left( {2x} \right)\\ &= \frac{{2x}}{{1 + {x^4}}}\end{aligned}\)

Thus, the derivative of the function y is \(\frac{{2x}}{{1 + {x^4}}}\).