The derivatives and the rules of differentiation limits

Steps to Solve

1. Understanding Derivatives The derivative of a function measures how the function value changes as its input changes. Mathematically, the derivative of a function $f(x)$ at a point $x$ is given by the limit:

$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$

2. Using the Power Rule For functions of the form $f(x) = x^n$, the derivative can be calculated using the power rule, which states:

$$ f'(x) = nx^{n-1} $$

3. Applying the Chain Rule If you have a composite function $f(g(x))$, the derivative can be found using the chain rule:

$$ f'(x) = f'(g(x)) \cdot g'(x) $$

4. Finding Limits To evaluate the limit for continuity of a function near a certain point $c$, we check:

$$ \lim_{x \to c} f(x) $$

This ensures that we analyze the function behavior as it approaches point $c$.

5. Combination of Rules In most calculus problems, you will need to combine these rules. For example, if you differentiate a product or quotient of two functions, use the product rule or quotient rule, respectively.