for which of the following functions is the chain rule an appropriate method to find the derivative with…

for which of the following functions is the chain rule an appropriate method to find the derivative with respect to x? i. y = sin(3x²) ii. y = e^x tan x iii. y = 1/(8x⁴ - 2x) a i only b ii only c iii only d i and iii only

for which of the following functions is the chain rule an appropriate method to find the derivative with respect to x? i. y = sin(3x²) ii. y = e^x tan x iii. y = 1/(8x⁴ - 2x) a i only b ii only c iii only d i and iii only

Answer

Explanation:

Step1: Recall chain - rule condition

The chain - rule is used for composite functions $y = f(g(x))$.

Step2: Analyze function I

For $y=\sin(3x^{2})$, let $u = 3x^{2}$, then $y=\sin(u)$. It is a composite function, so chain - rule can be used.

Step3: Analyze function II

For $y = e^{x}\tan x$, it is a product of two functions $e^{x}$ and $\tan x$. The product - rule $(uv)^\prime=u^\prime v + uv^\prime$ (where $u = e^{x}$ and $v=\tan x$) should be used, not the chain - rule.

Step4: Analyze function III

For $y=\frac{1}{8x^{4}-2x}=(8x^{4}-2x)^{-1}$, let $u = 8x^{4}-2x$, then $y = u^{-1}$. It is a composite function, so chain - rule can be used.

Answer:

D. I and III only