2. Consider the function $$ \mathrm{y}=\frac{1}{(3-\mathrm{x})^{4}} $$, find $$ \mathrm{f}^{\prime}(\mathrm{x}) $$. $$ \begin{array}{lll} ext { A. } \frac{4}{(3-x)^{5}} & ext { B. } \frac{(3-x)^{5}}{4} & ext { C. }(3 \ ext { 3. Which of the following is the derivative of } y=\ln \mathrm{x} ext { ? } \ \begin{array}{lll} ext { A. } \mathrm{x} & ext { B. }-x & ext { C. } \frac{1}{y}\end{array} \ ext { 4. What is } \frac{d}{d x}\left(\cos x^{2} ight) ext { ? } & \end{array} $$

Question

2. Consider the function $$ \mathrm{y}=\frac{1}{(3-\mathrm{x})^{4}} $$, find $$ \mathrm{f}^{\prime}(\mathrm{x}) $$. $$ \begin{array}{lll}	ext { A. } \frac{4}{(3-x)^{5}} & 	ext { B. } \frac{(3-x)^{5}}{4} & 	ext { C. }(3 \ 	ext { 3. Which of the following is the derivative of } y=\ln \mathrm{x} 	ext { ? } \ \begin{array}{lll}	ext { A. } \mathrm{x} & 	ext { B. }-x & 	ext { C. } \frac{1}{y}\end{array} \ 	ext { 4. What is } \frac{d}{d x}\left(\cos x^{2}ight) 	ext { ? } & \end{array} $$

UpStudy AI · Accepted Answer

**1. Derivative of $$y=\frac{1}{(3-x)^4}$$**

Write the function as  
$$
y = (3-x)^{-4}.
$$  
Differentiate using the chain rule:  
$$
\frac{dy}{dx} = -4(3-x)^{-5} \cdot \frac{d}{dx}(3-x).
$$  
Since  
$$
\frac{d}{dx}(3-x) = -1,
$$  
we have  
$$
\frac{dy}{dx} = -4(3-x)^{-5} \cdot (-1) = 4(3-x)^{-5} = \frac{4}{(3-x)^5}.
$$

---

**2. Derivative of $$y=\ln x$$**

Differentiate with respect to $$x$$:  
$$
\frac{dy}{dx} = \frac{1}{x}.
$$

---

**3. Derivative of $$y=\cos(x^2)$$**

Apply the chain rule. Let  
$$
u=x^2 \quad \text{so that} \quad y=\cos(u).
$$  
Differentiate:  
$$
\frac{dy}{du} = -\sin(u) \quad \text{and} \quad \frac{du}{dx} = 2x.
$$  
Then,  
$$
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} = -\sin(x^2) \cdot 2x = -2x\sin(x^2).
$$
**1. Derivative of $$ y = \frac{1}{(3 - x)^4} $$:** $$ f'(x) = \frac{4}{(3 - x)^5} $$ **Answer:** A. $$ \frac{4}{(3 - x)^5} $$ **2. Derivative of $$ y = \ln x $$:** $$ \frac{dy}{dx} = \frac{1}{x} $$ **Answer:** C. $$ \frac{1}{x} $$ **3. Derivative of $$ y = \cos(x^2) $$:** $$ \frac{dy}{dx} = -2x \sin(x^2) $$ **Answer:** Not listed among the options provided. **Note:** Option D is not applicable as it was not provided in the original question.

Consider the function , find .

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