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The task in the image involves approximating derivatives using difference quotients for various functions.

### 1. **Forward difference of** \( f(x) = \sin(x) \)
At \( x = -4 \) with step size \( h = \frac{1}{5} = 0.2 \), the forward difference formula is:

\[
f'(x) \approx \frac{f(x+h) - f(x)}{h}
\]
Substituting \( f(x) = \sin(x) \), \( x = -4 \), and \( h = 0.2 \):

\[
f'(x) \approx \frac{\sin(-4 + 0.2) - \sin(-4)}{0.2}
\]
This results in an approximation of \( -0.72 \).

### 2. **Backward difference of** \( f(x) = x^3 - x + 1 \)
At \( x = -4 \) with step size \( h = \frac{1}{10} = 0.1 \), the backward difference formula is:

\[
f'(x) \approx \frac{f(x) - f(x-h)}{h}
\]
Substituting \( f(x) = x^3 - x + 1 \), \( x = -4 \), and \( h = 0.1 \):

\[
f'(x) \approx \frac{(-4)^3 - (-4) + 1 - \left((-4 - 0.1)^3 - (-4 - 0.1) + 1\right)}{0.1}
\]
This gives an approximation of \( -580.99 \).

### 3. **Central difference of** \( f(x) = \ln(1 + x) \)
At \( x = 6 \) with step size \( h = \frac{1}{10} = 0.1 \), the central difference formula is:

\[
f'(x) \approx \frac{f(x+h) - f(x-h)}{2h}
\]
Substituting \( f(x) = \ln(1 + x) \), \( x = 6 \), and \( h = 0.1 \):

\[
f'(x) \approx \frac{\ln(1 + (6 + 0.1)) - \ln(1 + (6 - 0.1))}{2(0.1)}
\]
This results in an approximation of \( 0.95 \).

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Would you like more details on how these approximations were calculated? Here are 5 related questions:

1. How is the forward difference approximation derived?
2. What are the benefits of using the central difference over forward or backward differences?
3. How does the choice of \( h \) affect the accuracy of these approximations?
4. How can you calculate higher-order derivatives using difference methods?
5. What other methods exist for approximating derivatives?

**Tip:** Smaller step sizes \( h \) tend to give more accurate approximations, but very small \( h \) can introduce numerical errors due to rounding.

Learn how to approximate the derivatives of various functions such as trigonometric, polynomial, and logarithmic using difference quotients. This tutorial includes the use of forward, backward, and central difference methods to estimate derivatives at specific points.

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