Which of the following rules don't you have to use to differentiate the function $$\frac{\sin(x)+\cos(x)\ln(\tan(x))}{e^x}$$

Question

BlackTom AI · Accepted Answer

To differentiate the given function, we should first restate the problem clearly and then assess each rule in turn.
Option a) Chain Rule: (f(g(x)))' = f'(g(x)) · g'(x)
- This rule is generally needed when you have composite functions, such as sin x, ln(tan x), or any inner functions. In our target function, ln(tan(x)) contains a chain of tan inside the natural log, and sin(x) and cos(x) are composed with x. Therefore, chain rule is a natural and typically necessary tool to handle these inner layers.
Option b) Addition Rule: (f(x) + g(x))' = f'(x) + g'(x)
- The numerator sin(x) + cos(x) ln(tan(x)) is a sum of two functions. Differentiating a sum requires the sum rule. So this rule is needed to differentiate the numerator term-by-term.
Option c) Quotient Rule: (f(x)/g(x))' = [f'(x)g(x) − f(x)g'(x)]/[g(x)]^2
- One standard route is to treat the whole expression as a quotient and apply the quotient rule. If you take this approach, you would use the quotient rule to differentiate. However, you could alternatively rewrite the original expression as a product to avoid the quotient rule entirely.
Option d) Power Rule: (f(x)^{g(x)})' = f(x)^{g(x)} [ g'(x) ln(f(x)) + (g(x) f'(x))/f(x) ]
- This is a specialized rule for differentiating functions where the base and exponent are both functions of x. Our given function does not present a variable-base raised to a variable-exponent form. The entire expression is a quotient (or a product after rewriting) of simple functions, not a function of the type f(x)^{g(x)}. Thus, this Power Rule is not required for differentiating this particular expression.
Option e) Product Rule: (f(x)g(x))' = f'(x)g(x) + f(x)g'(x)
- If we rewrite the original quotient as a product, namely (sin(x) + cos(x) ln(tan(x))) · e^{−x}, we must differentiate a product of two functions. In that pathway, the product rule becomes essential.

In summary, chain rule, addition rule, quotient rule (depending on the chosen method), and product rule are all relevant to the differentiation process depending on the approach taken. The Power Rule for variable bases and exponents is not required for this problem, making option d the rule that isn't needed in differentiating the given function.

MTH1030 -1035 - S1 2025 MTH1030/35 Week 11 lesson quiz: Differential equations

Which of the following rules don't you have to use to differentiate the function \[\frac{\sin(x)+\cos(x)\ln(\tan(x))}{e^x}\]

查看解析

登录即可查看完整答案

类似问题

Given the following functions, their graphs and the composite functions represented graphically, verify they are correct representations using derivative rules. Consider the functions: f(x) = 3x^2 + 2x + 1 g(x) = 4x^3 - x +5 (I:1)

Assume y=f(x) and y=g(x) are non-constant functions. Let a and c be constants. Select the appropriate derivative rule to be used for each. d dx (ca) Constant Rule d dx (ag(x)) Exponential Rule d dx ((f(x))c) Power Rule d dx ((f(x))g(x)) Logarithmic Differentiation

In a consumer society, many adults channel creativity into buying things

Economic stress and unpredictable times have resulted in a booming industry for self-help products

People born without creativity never can develop it

A product has a selling price of $20, a contribution margin ratio of 40% and fixed cost of $120,000. To make a profit of $30,000. The number of units that must be sold is: Type the number without $ and a comma. Eg: 20000

Which of the following statement regarding cost is correct:

Under the assumptions used in cost-volume-profit analysis, as the activity increases:

更多留学生实用工具

考试浏览器助手

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