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多重下拉选择题

类似问题

Question text 3Marks Differentiate the following function at x=1 using the chain rule: [math: f(x)=(4x3−4x2+6x−7)5]. The derivative is [math: f′(1)=] Answer 1[input].Notes Report question issue Question 5 Notes

Which of these is the result when you differentiate the expression ln ⁡ ( 3 cos ⁡ ( 𝑥 ) ) with respect to 𝑥 ?

Differentiate 5 sin ⁡ ( 𝑥 3 + 2 𝑥 )

Which rule would be used to differentiate the function

Find dy/dt. y = cos()

If f ( x ) = 5 x − 1 , g ( 0 ) = 2 , g ′ ( 0 ) = 4 , and h ( x ) = f ( g ( x ) ) {"version":"1.1","math":"f(x)=5x-1,\ g(0)=2,\ g'(0)=4, \text{ and } h(x)=f(g(x))"}then what is the value of h ′ ( 0 ) ? {"version":"1.1","math":"h'(0)?"}

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单