Question

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

Derivatives
ANSWERED
asked 2021-05-31
Use the Chain Rule to calculate the derivatives of the following functions.
\(\displaystyle{y}={\cos{{5}}}{t}\)

Answers (1)

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

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