91Ó°ÊÓ

Differentiable functions form the heart of calculus, enabling us to find rates of change and solve real-world problems. But what exactly does it mean for a function to be differentiable? Well, if a function is smooth enough without any breaks, jumps, or sharp corners, it is considered differentiable.
Differentiability is defined as having a derivative at every point in its domain. This means the function must have a continuous rate of change. More formally, a function \( f(x) \) is differentiable at a point \( a \) if the following limit exists:

• \( \lim_{h \to 0} \frac{f(a+h) - f(a)}{h} \).

If this limit exists, then \( f'(a) \) is the derivative of \( f \) at point \( a \). Differentiable functions help us apply calculus tools like the Product Rule, Chain Rule, and more, to analyze complex problems.

The concept of a derivative is foundational in calculus. It represents the rate at which a function's value changes as its input changes. In simple terms, the derivative of a function gives us the slope of the tangent line to the function's graph at any point.
Mathematically, the derivative of a function \( f(x) \) at a point \( x \) is denoted as \( f'(x) \), and it can be calculated using the limit:

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

There are specific rules to compute derivatives, such as the Product Rule, which we saw in the exercise. This rule is crucial when dealing with the product of two or more differentiable functions. If \( u(x) \) and \( v(x) \) are differentiable functions, then their product's derivative is \( (uv)' = u'v + uv' \). Understanding and mastering the derivative concept is key to solving many practical problems involving rates of change.

The Chain Rule is a powerful tool for differentiating composite functions. Whenever we have a function nested inside another function, the Chain Rule comes into play. In essence, it helps us find the derivative of a composite function by taking the derivative of the outer function and multiplying it by the derivative of the inner function.
The Chain Rule can be written as:

• Given \( y = f(g(x)) \, \text{the derivative is} \quad \frac{dy}{dx} = f'(g(x)) \cdot g'(x) \).

This rule allows us to efficiently handle derivations that involve complex, multi-layered functions. For example, differentiating \( y = e^{3x} \) requires applying the Chain Rule. The outer function is \( e^{u} \), where \( u = 3x \). We derive the outer function, \( e^{u} \, \) and then multiply it by the derivative of the inner function, \( 3x \. \) Therefore, the derivative is \( 3e^{3x} \. \) Mastery of the Chain Rule is essential for tackling many differentiation problems encountered in calculus.