Question

# Use the Chain Rule to calculate the derivatives of the following functions. y=\cos 5t

Derivatives
Use the Chain Rule to calculate the derivatives of the following functions.
$$\displaystyle{y}={\cos{{5}}}{t}$$

2021-06-01

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 $$\displaystyle{y}={\cos{{5}}}{t}\ {i}{s}\ −{5}{\sin{{5}}}{t}$$.