For the given composite function, identify the inner function, 
u = g(x),
 and the outer function, 
y = f(u).
 (Use non-identity functions for 
f(x)
 and 
g(x).)
y = 
3

1 + 7x
 
(f(u), g(x)) = 

 
  

Find the derivative 
dy
dx
.
dy
dx
 = 
 

Let's start by identifying the **inner function** \( g(x) \) and the **outer function** \( f(u) \) from the given composite function.

The given function is:

\[
y = \frac{3}{1 + 7x}
\]

### Step 1: Identify the inner function
We can see that inside the denominator is the expression \( 1 + 7x \). Therefore, we set this as the inner function:

\[
g(x) = 1 + 7x
\]

### Step 2: Identify the outer function
Now, if we treat \( g(x) = 1 + 7x \) as the variable \( u \), then we rewrite the given function as:

\[
y = \frac{3}{u}
\]

So, the outer function is:

\[
f(u) = \frac{3}{u}
\]

Thus, the composite function is:

\[
y = f(g(x)) = \frac{3}{1 + 7x}
\]

### Step 3: Find the derivative \( \frac{dy}{dx} \)

Using the chain rule:

\[
\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}
\]

- First, find \( \frac{dy}{du} \). Since \( f(u) = \frac{3}{u} \), the derivative of \( f(u) \) with respect to \( u \) is:

\[
\frac{dy}{du} = \frac{d}{du} \left( \frac{3}{u} \right) = -\frac{3}{u^2}
\]

- Next, find \( \frac{du}{dx} \). Since \( g(x) = 1 + 7x \), the derivative of \( g(x) \) with respect to \( x \) is:

\[
\frac{du}{dx} = 7
\]

### Step 4: Apply the chain rule
Now, multiply the derivatives:

\[
\frac{dy}{dx} = \left( -\frac{3}{u^2} \right) \cdot 7
\]

Since \( u = 1 + 7x \), substitute back:

\[
\frac{dy}{dx} = -\frac{3 \cdot 7}{(1 + 7x)^2}
\]

\[
\frac{dy}{dx} = -\frac{21}{(1 + 7x)^2}
\]

### Final Answer:
\[
\frac{dy}{dx} = -\frac{21}{(1 + 7x)^2}
\]

Would you like further details or have any questions?

Here are five related questions you can explore:
1. What is the significance of the chain rule in calculus?
2. How do you apply the product rule in derivatives?
3. Can you explain how to differentiate more complex composite functions?
4. How would you interpret the derivative \( \frac{dy}{dx} \) graphically?
5. How do implicit differentiation and chain rule differ in usage?

**Tip:** When applying the chain rule, always identify the inner and outer functions carefully, as they guide the differentiation process!

For the given composite function, identify the inner function, u = g(x), and the outer function, y = f(u). (Use non-identity functions for f(x) and g(x).) y = 3 / (1 + 7x) (f(u), g(x)) = ? Find the derivative dy/dx.

Learn how to find the derivative of the composite function y = 3 / (1 + 7x) using the chain rule. This detailed solution identifies the inner and outer functions and applies calculus techniques to compute the derivative. Ideal for students in Grades 11-12 studying advanced calculus.

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Find the derivative dy dx . dy dx

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