How do you differentiate #f(x)=tan(e^(1/x)) # using the chain rule?

Question

How do you differentiate #f(x)=tan(e^(1/x))   # using the chain rule?

Luke Alvarez · Accepted Answer

#d/dx[f(g(x))]=f'(g(x))g'(x)#

The chain rule allows us to take the derivative of a composition of two or more functions. For two functions:

#d/dx[f(g(x))]=f'(g(x))g'(x)#

We can see that the function #f(x)# is composed of three functions, each nestled inside the last.

By the chain rule, we first need to take the derivative of the outermost function and work our way in. It's like taking the function apart layer by layer until we get all the way through.

First, we take the derivative of the tangent function, which is #sec^2#. This gives us:

#sec^2(e^(1/x))#

Next, we take the derivative of the #e# term. This, as always, is just itself. We multiply this by the first portion of the derivative as we determined above. Thus, we have so far:

#sec^2(e^(1/x))*e^(1/x)#

Lastly, we take the derivative of #1/x#. Recognizing that this is equivalent to #x^-1#, we have #-x^(-2)#. And so, our final answer becomes:

#-sec^2(e^(1/x))e^(1/x)x^-2#

This can also be written as:

#(-sec^2(e^(1/x))e^(1/x))/x^2#

Ezekiel Alexander · Answer

undefined To differentiate $ f(x) = \tan(e^{1/x}) $ using the chain rule, follow these steps:

1. Identify the outer function and the inner function. In this case, the outer function is $ \tan(x) $ and the inner function is $ e^{1/x} $.
2. Compute the derivative of the outer function with respect to its argument. The derivative of $ \tan(x) $ is $ \sec^2(x) $.
3. Compute the derivative of the inner function with respect to $ x $. The derivative of $ e^{1/x} $ can be found using the chain rule as follows: $ \frac{d}{dx}(e^{1/x}) = e^{1/x} \cdot (-1/x^2) $.
4. Substitute the inner function into the derivative of the outer function.
5. Multiply the results from steps 2 and 3 to get the derivative of the composite function $ f(x) $.

Therefore, the derivative of $ f(x) = \tan(e^{1/x}) $ is:

$$ f'(x) = \sec^2(e^{1/x}) \cdot e^{1/x} \cdot (-1/x^2) $$