If wanting to differentiate:

𝑦
=
1
𝑥
2
+
1

The most appropriate rule to use would be the
[ Select ]
chain
general
right handed
product
rule

Question

If wanting to differentiate:

𝑦
=
1
𝑥
2
+
1

The most appropriate rule to use would be the 
[ Select ]
chain
general
right handed
product
 rule

BlackTom AI · Accepted Answer

We are differentiating the function y = 1/x^2 + 1 with respect to x. 
First, notice that the function is a sum of two terms: 1/x^2 and 1, where 1 is a constant and differentiates to 0. 
For the term 1/x^2, rewrite it as x^(-2). Then, apply the power rule: d/dx [x^n] = n*x^(n-1). Here, n = -2, so d/dx [x^(-2)] = -2*x^(-3) = -2/x^3. 
Thus, the derivative of the entire function is dy/dx = -2/x^3 + 0 = -2/x^3. 
Now consider which differentiation rule is appropriate for obtaining this result. 
- Chain rule: Used when differentiating composite functions, such as f(g(x)). In this expression, there is no inner function being composed inside x^(-2) beyond the simple power, so the chain rule is not required explicitly for the differentiation step beyond the inner variable power transformation we performed. 
- Product rule: Applies when differentiating a product of two functions, i.e., d/dx [u(x)v(x)] = u'v + uv'. Our function is not given as a product of two variable functions, so the product rule is not applicable here. 
- Quotient rule/general rule: While 1/x^2 resembles a quotient, it is more straightforward to rewrite as x^(-2) and use the power rule directly; the quotient rule is not necessary for this simple reciprocal power form, and claiming a separate general rule would be an overcomplication. 
- Right-handed rule: This is not a standard differentiation rule in calculus; it does not pertain to the differentiation process and is not applicable. 
Therefore, the most appropriate approach is to treat 1/x^2 as x^(-2) and apply the power rule directly, with the understanding that the chain rule is not invoked in a case needing no inner function substitution. 
If you were to encounter a more complex composite like sin(3x^2), you would then invoke the chain rule to handle the inner function 3x^2. In this simpler expression, the chain rule is not required beyond the basic power differentiation, and the product and quotient rules are unnecessary for this particular form.

If wanting to differentiate: 𝑦 = 1 𝑥 2 + 1 The most appropriate rule to use would be the [ Select ] chain general right handed product rule

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