Question

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

63. \(f\left( x \right) = {\bf{si}}{{\bf{n}}^{ - {\bf{1}}}}\left( {{\bf{5}}x} \right)\)

Short answer

The derivative of the function \(f\left( x \right)\) is \(\frac{5}{{\sqrt {1 - 25{x^2}} }}\).

Step-by-step solution

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

The differentiation of \({\sin ^{ - 1}}x\) with respect to xis:

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

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

The differentiation of \(f\left( x \right)\) with respect to xis:

\(\begin{aligned}f'\left( x \right) &= \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {{{\sin }^{ - 1}}\left( {5x} \right)} \right)\\ &= \frac{1}{{\sqrt {1 - {{\left( {5x} \right)}^2}} }} \cdot \frac{{\rm{d}}}{{{\rm{d}}x}}\left( {5x} \right)\\ &= \frac{1}{{\sqrt {1 - 25{x^2}} }}\left( 5 \right)\\ &= \frac{5}{{\sqrt {1 - 25{x^2}} }}\end{aligned}\)

Thus, the derivative of the function \(f\left( x \right)\) is \(\frac{5}{{\sqrt {1 - 25{x^2}} }}\).