The image contains a mathematical expression and a space for its derivative. It shows:\[ f(w) = \ln(\cos(w - 5)) \]Below it, there is an empty box labeled:\[ f'(w) = \]This setup is intended for users to calculate and input the derivative of the given function $ f(w) = \ln(\cos(w - 5)) $.

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Description: The image contains a mathematical expression and a space for its derivative. It shows:\[ f(w) = \ln(\cos(w - 5)) \]Below it, there is an empty box labeled:\[ f'(w) = \]This setup is intended for users to calculate and input the derivative of the giv

Answered: f(w) = In(cos(w– 5)) f'(w) =

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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The image contains a mathematical expression and a space for its derivative. It shows:

\[ f(w) = \ln(\cos(w - 5)) \]

Below it, there is an empty box labeled:

\[ f'(w) = \]

This setup is intended for users to calculate and input the derivative of the given function $ f(w) = \ln(\cos(w - 5)) $.

Expert Solution

Step 1

The given function f(w) is,

$f (w) = \ln (\cos (w - 5))$

The derivative of the function f(w) is represented by f'(w).

The given function contains two functions, thus during differentiation after differentiating the main function (logarithmic function), the result is multiplied with the differential of the inner function (cosine function)

Thus, the derivative or f'(w) of the function by is given by:

$\begin{array}{rcl} f' (w) & = & \frac{d f (w)}{d w} \\ = & \frac{d (\ln (\cos (w - 5))}{d w} \\ = & \frac{1}{\cos (w - 5)} \times \frac{d c o s (w - 5)}{d w} \\ = & \frac{1}{\cos (w - 5)} \times - \sin (w - 5) \\ = & - \tan (w - 5) \end{array}$

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