QUESTION 35 Use the chain rule to find the derivative of $$ y=(2 x+1)^{7} $$

Question

UpStudy AI · Accepted Answer

To find the derivative of $$ y = (2x + 1)^7 $$ using the **chain rule**, follow these steps:

### **Step 1: Identify the Outer and Inner Functions**

The **chain rule** is used to differentiate composite functions. A composite function is composed of an outer function and an inner function.

- **Outer Function**: $$ f(u) = u^7 $$
- **Inner Function**: $$ u = 2x + 1 $$

So, $$ y = f(u) = (2x + 1)^7 $$.

### **Step 2: Differentiate the Outer Function with Respect to the Inner Function**

Differentiate $$ f(u) = u^7 $$ with respect to $$ u $$:

$$
\frac{df}{du} = 7u^6
$$

### **Step 3: Differentiate the Inner Function with Respect to $$ x $$**

Differentiate $$ u = 2x + 1 $$ with respect to $$ x $$:

$$
\frac{du}{dx} = 2
$$

### **Step 4: Apply the Chain Rule**

The **chain rule** states that:

$$
\frac{dy}{dx} = \frac{df}{du} \cdot \frac{du}{dx}
$$

Substitute the derivatives we've found:

$$
\frac{dy}{dx} = 7u^6 \cdot 2 = 14u^6
$$

### **Step 5: Substitute Back the Inner Function**

Replace $$ u $$ with $$ 2x + 1 $$:

$$
\frac{dy}{dx} = 14(2x + 1)^6
$$

### **Final Answer**

$$
\boxed{\frac{dy}{dx} = 14\, (2x + 1)^6}
$$
The derivative of $$ y = (2x + 1)^7 $$ is $$ \frac{dy}{dx} = 14(2x + 1)^6 $$.

QUESTION 35 Use the chain rule to find the derivative of \( y=(2 x+1)^{7} \)