Question

Find the derivative of the function:

29. \(r\left( t \right) = {10^{2\sqrt t }}\)

Short answer

The derivative of \(r\left( t \right)\) is \(\frac{{\left( {\ln 10} \right){{10}^{2\sqrt t }}}}{{\sqrt t }}\).

Step-by-step solution

The chain rule

If \(g\) is differentiable at \(x\) and \(f\) is differentiable at \(g\left( x \right)\), then the composite function \(F = f \circ g\) defined by \(F\left( x \right) = f\left( {g\left( x \right)} \right)\) is differentiable at \(x\) and \(F'\) is given by the product \(F'\left( x \right) = f'\left( {g\left( x \right)} \right) \cdot g'\left( x \right)\). In Leibniz notation, if \(y = f\left( u \right)\) and \(u = g\left( x \right)\) are both differentiable functions, then \(\frac{{dy}}{{dx}} = \frac{{dy}}{{du}} \cdot \frac{{du}}{{dx}}\).

The derivative of the given function

Differentiate \(r\left( t \right)\) with respect to \(t\) as follows:

\(\begin{aligned}f'\left( t \right) &= \frac{d}{{dt}}\left( {{{10}^{2\sqrt t }}} \right)\\ &= {10^{2\sqrt t }}\ln 10\frac{d}{{dt}}\left( {2\sqrt t } \right)\\ &= {10^{2\sqrt t }}\ln 10\left( {2\left( {\frac{1}{{2\sqrt t }}} \right)} \right)\\ &= {10^{2\sqrt t }}\ln 10\left( {\frac{1}{{\sqrt t }}} \right)\\ &= \frac{{\left( {\ln 10} \right){{10}^{2\sqrt t }}}}{{\sqrt t }}\end{aligned}\)

Hence, the derivative of \(r\left( t \right)\) is \(\frac{{\left( {\ln 10} \right){{10}^{2\sqrt t }}}}{{\sqrt t }}\).