Why Chain Rule for Differentiating f(u) = e1/u?

In summary, the function f(u) = e1/u requires the use of the chain rule for differentiation. This is because if we assume x = 1/u, then the function becomes f(x(u)) and we have to use the chain rule to find the derivative. The exponent in the function can be represented by any variable, it does not have to be "x" specifically.
  • #1
merced
44
1
Differentiate the function: f(u) = e1/u
So, I used the chain rule and figured out that
f '(u) = (-u-2) e1/u

My question is, why do you have to use the chain rule?
I know that if f(x) = ex
then f '(x) = ex

Why can't I pretend that 1/u is x and then say that
f '(x) = ex = e1/u

In other words, does the exponent always have to be "x" only, for f '(x) = ex to work?
 
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  • #2
merced said:
Differentiate the function: f(u) = e1/u
So, I used the chain rule and figured out that
f '(u) = (-u-2) e1/u

My question is, why do you have to use the chain rule?
I know that if f(x) = ex
then f '(x) = ex

Why can't I pretend that 1/u is x and then say that
f '(x) = ex = e1/u

In other words, does the exponent always have to be "x" only, for f '(x) = ex to work?
The derivative e^u with respect to u is e^u and the derivative e^x with respect to x is e^x, and it does not matter what alpahbet you choose to denote the variable with. It's a dummy.

But in the problem you have posted, if you assume that x = 1/u, then the function is f(x(u)) [since x is now a function of u], and that is why you use the chain rule. You assume it to be a function of a function. Therefore [tex]\frac{df}{du} = \frac{df}{dx}\frac{dx}{du}[/tex]
 
  • #3
Oooh, ok, thanks
 

FAQ: Why Chain Rule for Differentiating f(u) = e1/u?

What is the chain rule for differentiating f(u) = e1/u?

The chain rule is a rule in calculus that allows you to find the derivative of a function that is composed of two or more other functions. In the case of f(u) = e1/u, the chain rule states that the derivative is equal to the derivative of the outer function (ex) evaluated at the inner function (1/u), multiplied by the derivative of the inner function.

How do you apply the chain rule to differentiate f(u) = e1/u?

To apply the chain rule, you first identify the inner and outer functions. In this case, the outer function is ex and the inner function is 1/u. Next, you find the derivatives of each function, which are ex and -1/u2, respectively. Finally, you plug these values into the chain rule formula, resulting in the derivative of f(u) = e1/u being -e1/u / u2.

Why is the chain rule important for differentiating f(u) = e1/u?

The chain rule is important because it allows us to find the derivative of more complex functions that are composed of multiple simpler functions. Without the chain rule, it would be much more difficult to find the derivative of functions like f(u) = e1/u.

Can the chain rule be applied to functions other than f(u) = e1/u?

Yes, the chain rule can be applied to any function that is composed of two or more other functions. This includes functions with different bases, such as f(u) = 21/u, as well as functions with more complex compositions, such as f(u) = sin(cos(u)).

What are some common mistakes when using the chain rule to differentiate f(u) = e1/u?

Some common mistakes when using the chain rule include forgetting to multiply by the derivative of the inner function, plugging in the wrong values for the derivatives of the inner and outer functions, and not using the correct chain rule formula. It is important to carefully identify the inner and outer functions and correctly apply the chain rule formula to avoid these mistakes.

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