More Derivative Examples

1. Derivatives for Non-Linear Functions

In previous examples, we considered linear functions where the slope was constant.
However, for non-linear functions, the slope (derivative) can vary depending on the input value.

This means:

The derivative is not always constant
It depends on the point at which it is evaluated

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2. Example 1: $f(a) = a^2$

Case 1: $a = 2$

• $f(2) = 4$

If we increase $a$ slightly:

• $a = 2.001$
• $f(a) \approx 4.004$

Observation:

• Change in input: $0.001$
• Change in output: $0.004$

$$\text{Slope} = \frac{0.004}{0.001} = 4$$

So:$$\frac{d}{da} f(a) = 4 \quad \text{when } a = 2$$

Case 2: $a = 5$

• $f(5) = 25$

If we increase $a$ slightly:

• $a = 5.001$
• $f(a) \approx 25.010$

Observation:

• Change in output: $0.010$

$$\text{Slope} = \frac{0.010}{0.001} = 10$$

So:$$\frac{d}{da} f(a) = 10 \quad \text{when } a = 5$$

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3. General Derivative Formula

From calculus, the derivative of $f(a) = a^2$ is:

$$\frac{d}{da}(a^2) = 2a$$

This matches our observations:

• At $a = 2$: $2a = 4$
• At $a = 5$: $2a = 10$

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4. Key Insight

For non-linear functions:

The slope changes depending on the value of aaa

This is different from linear functions, where the slope is constant everywhere.

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5. Example 2: $f(a) = a^3$

From calculus:

$$\frac{d}{da}(a^3) = 3a^2$$

Case: $a = 2$

• $f(2) = 8$

If $a$a increases slightly:

• $f(a) \approx 8.012$

This corresponds to:

$$3a^2 = 3 \times 4 = 12$$

The output increases 12 times faster than the input change.

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6. Example 3: $f(a) = \log(a)$

Using natural logarithm:

$$\frac{d}{da} \log(a) = \frac{1}{a}$$

Case: $a = 2$

• Small increase in $a$: $0.001$
• Expected increase in $f(a)$:

$$\frac{1}{2} \times 0.001 = 0.0005$$

This matches the observed change.

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7. Approximation vs Exact Definition

In these examples, we used small changes like $0.001$.

However, the formal definition of a derivative uses:

infinitesimally small changes (approaching zero)

This ensures that the derivative gives an exact slope.

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8. Summary of Key Patterns

• $f(a) = a^2$ → slope depends on $a$
• $f(a) = a^3$ → slope grows faster
• $f(a) = \log(a)$ → slope decreases as $a$ increases

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9. Key Takeaways

1. A derivative represents the slope of a function
2. For non-linear functions, the slope varies across different points
3. Derivative formulas allow us to compute slopes efficiently
4. Small changes in input lead to predictable changes in output based on the derivative