https://file.aiquickdraw.com/mathbot/file/1726814766325kx2nhn4c.jpg

Solve the problem and return an answer choice

We are given the graphs of two functions \( f(x) \) and \( g(x) \), and the function \( h(x) = \frac{f(x) + 4}{g(x) + 2x} \). We are asked to find \( h'(3) \), the derivative of \( h(x) \) at \( x = 3 \).

### Step 1: Differentiate \( h(x) \)
We will use the quotient rule to differentiate \( h(x) \). The quotient rule states that for a function of the form \( \frac{u(x)}{v(x)} \), the derivative is given by:
\[
h'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}
\]
In this case, \( u(x) = f(x) + 4 \) and \( v(x) = g(x) + 2x \).

Now, let's find the derivatives:
- \( u'(x) = f'(x) \) because the derivative of the constant 4 is 0.
- \( v'(x) = g'(x) + 2 \) because the derivative of \( 2x \) is 2.

Thus, applying the quotient rule:
\[
h'(x) = \frac{(f'(x))(g(x) + 2x) - (f(x) + 4)(g'(x) + 2)}{(g(x) + 2x)^2}
\]

### Step 2: Calculate values at \( x = 3 \)
We need to find the values of \( f(3) \), \( f'(3) \), \( g(3) \), and \( g'(3) \) using the graphs.

- From the graph of \( f(x) \), we can see that \( f(3) = 5 \).
- The slope of \( f(x) \) from \( x = 2 \) to \( x = 4 \) is:
  \[
  f'(3) = \frac{f(4) - f(2)}{4 - 2} = \frac{3 - 4}{2} = \frac{-1}{2}
  \]

- From the graph of \( g(x) \), we observe that \( g(3) = 5 \).
- The slope of \( g(x) \) from \( x = 2 \) to \( x = 4 \) is:
  \[
  g'(3) = \frac{g(4) - g(2)}{4 - 2} = \frac{1 - 5}{2} = \frac{-4}{2} = -2
  \]

### Step 3: Plug values into the derivative formula
Now substitute \( x = 3 \), along with the values we calculated, into the formula for \( h'(x) \):
\[
h'(3) = \frac{\left( \frac{-1}{2} \right)(5 + 6) - (5 + 4)(-2 + 2)}{(5 + 6)^2}
\]
Simplifying the terms:
\[
h'(3) = \frac{\left( \frac{-1}{2} \right)(11) - (9)(0)}{11^2} = \frac{\frac{-11}{2}}{121} = \frac{-11}{2 \times 121} = \frac{-11}{242}
\]

Thus:
\[
h'(3) = \frac{-1}{22}
\]

This does not match the available answer choices. Upon review, let's check if there was an arithmetic mistake.

The correct differentiation

The graphs of the functions f and g are shown above. If h(x) = \frac{f(x)+4}{g(x)+2x}, then h'(3) = 

Solve the calculus problem involving differentiation of the function h(x) = \frac{f(x)+4}{g(x)+2x} at x = 3 using the quotient rule. The solution includes step-by-step differentiation and evaluation using the given graphs of functions f and g. Suitable for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation