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

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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 \[\frac{\sin(x)+\cos(x)\ln(\tan(x))}{e^x}\]

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