Step 1

Given \(\displaystyle{y}={\cos{{5}}}{t}\)

To use The Chain Rule to calculate the derivatives of the above function.

Identity Used \(\displaystyle{\frac{{{d}}}{{{\left.{d}{x}\right.}}}}{\left({f{{\left({g{{\left({x}\right)}}}\right)}}}\right)}={f}'{\left({g{{\left({x}\right)}}}\right)}\cdot{g}'{\left({x}\right)}\),

Step 2

Explanation- Rewrite the given expression,

\(\displaystyle{y}={\cos{{5}}}{t}\)

As per the chain rule of derivative , solving as follows,

\(\displaystyle{\frac{{{\left.{d}{y}\right.}}}{{{\left.{d}{t}\right.}}}}=-{5}{\sin{{t}}}\cdot{\frac{{{d}}}{{{\left.{d}{t}\right.}}}}{\left({5}{t}\right)}\)

\(\displaystyle=-{\sin{{5}}}{t}\cdot{5}\)

\(\displaystyle=-{5}{\sin{{5}}}{t}\)

So, the derivative of the expression \(y=\cos 5t\ is\ −5 \sin 5t\).

Answer- the derivative of the expression \(\displaystyle{y}={\cos{{5}}}{t}\ {i}{s}\ −{5}{\sin{{5}}}{t}\).