Questions: Consider the following function and express the relationship between a small change in x and the corresponding change in y in the form dy=f'(x) dx. f(x)=cot 8 x dy=

Consider the following function and express the relationship between a small change in x and the corresponding change in y in the form dy=f'(x) dx.
f(x)=cot 8 x
dy=

Transcript text: Consider the following function and express the relationship between a small change in x and the corresponding change in y in the form $\mathrm{dy}=\mathrm{f}^{\prime}(\mathrm{x}) \mathrm{dx}$. \[ f(x)=\cot 8 x \] \[ d y= \]

Solution

Solution Steps

Step 1: Identify the function and its derivative

The given function is: \[ f(x) = \cot(8x) \] To find the relationship between a small change in $ x $ and the corresponding change in $ y $, we need to compute the derivative of $ f(x) $ with respect to $ x $.

Step 2: Compute the derivative of $ f(x) $

The derivative of $ \cot(u) $ with respect to $ u $ is $ -\csc^2(u) $. Using the chain rule, the derivative of $ f(x) = \cot(8x) $ is: \[ f'(x) = -\csc^2(8x) \cdot \frac{d}{dx}(8x) \] \[ f'(x) = -\csc^2(8x) \cdot 8 \] \[ f'(x) = -8\csc^2(8x) \]

Step 3: Express the relationship between $ dy $ and $ dx $

The relationship between a small change in $ x $ and the corresponding change in $ y $ is given by: \[ dy = f'(x) \, dx \] Substituting the derivative $ f'(x) $ we found: \[ dy = -8\csc^2(8x) \, dx \]

Final Answer

\[ \boxed{dy = -8\csc^2(8x) \, dx} \]

Was this solution helpful?

Unhelpful

Helpful

Questions asked by other users

< Previous Next >

Thank you for your feedback.

Copied!