How do you use the chain rule to differentiate #f(x)=sin(tan(5+1/x)-7x)#?

Question

Samantha Adkison · Accepted Answer

See the answer below:

Luke Alvarez · Answer

undefined To differentiate $ f(x) = \sin(\tan(5 + 1/x) - 7x) $ using the chain rule, follow these steps:

1. Identify the outer function and the inner function.
2. Differentiate the outer function with respect to the inner function.
3. Multiply the result by the derivative of the inner function with respect to $ x $.

Here's the breakdown:

1. The outer function is $ \sin(u) $, where $ u = \tan(5 + 1/x) - 7x $.
   The inner function is $ u = \tan(v) $, where $ v = 5 + 1/x $.
2. Differentiate the outer function: $ \frac{d}{du}(\sin(u)) = \cos(u) $.
3. Differentiate the inner function: 
   $ \frac{dv}{dx} = \frac{d}{dx}(5 + 1/x) = 0 - \frac{1}{x^2} = -\frac{1}{x^2} $.
   $ \frac{du}{dv} = \frac{d}{dv}(\tan(v)) = \sec^2(v) = \sec^2(5 + 1/x) $.
4. Apply the chain rule:
   $ \frac{d}{dx}(\sin(\tan(5 + 1/x) - 7x)) = \cos(\tan(5 + 1/x) - 7x) \cdot (-\frac{1}{x^2}) \cdot \sec^2(5 + 1/x) $.

So, the derivative of $ f(x) $ is:

$$ f'(x) = \cos(\tan(5 + 1/x) - 7x) \cdot (-\frac{1}{x^2}) \cdot \sec^2(5 + 1/x) $$