Differentiability
To differentiate a function, it must be differentiable. This means that the function has a well-defined slope at every point. Not all functions satisfy this — some have jumps, cusps, or corners where the derivative does not exist.
When is a function differentiable
A function \( \large f(x) \) is differentiable at a point \( \large x_0 \) if the derivative exists there, that is, if the limit
$$ \large f'(x_0) = \lim_{h \to 0} \frac{f(x_0 + h) - f(x_0)}{h} $$
gives the same result whether one approaches from the left or from the right. If the left- and right-hand slopes are not equal, there is no tangent — and hence no derivative.
A function that is differentiable throughout its domain is called smooth. All polynomials and exponential functions are smooth, while for example the absolute-value function \( \large f(x) = |x| \) is not, because it has a corner at the origin.
Relation between continuity and differentiability
If a function is differentiable, it is also continuous — but the converse is not necessarily true. It is therefore possible to have a function that is smooth enough to be drawn without breaks, yet still not differentiable everywhere. A classic example is \( \large f(x) = |x| \): it is continuous but not differentiable at \( \large x = 0 \).
Rules of differentiation
When finding derivatives, it is rarely necessary to return to the definition. Instead, one uses a set of fixed differentiation rules that make computation quick and systematic. These rules apply to all differentiable functions.
1. Constant rule
The derivative of a constant is always zero:
$$ \large (k)' = 0 $$
Example: If \( \large f(x) = 7 \), then \( \large f'(x) = 0 \).
2. Power rule
The most important rule is the power rule:
$$ \large (x^n)' = n \cdot x^{n-1} $$
Example: \( \large (x^4)' = 4x^3 \) and \( \large (x^{1/2})' = \frac{1}{2}x^{-1/2} \).
3. Constant factor rule
A constant factor can be taken outside the derivative:
$$ \large (k \cdot f(x))' = k \cdot f'(x) $$
Example: \( \large (3x^2)' = 3 \cdot 2x = 6x \).
4. Sum and difference rule
The derivative of a sum (or difference) is the sum (or difference) of the derivatives:
$$ \large (f(x) + g(x))' = f'(x) + g'(x) $$
Example: \( \large (x^3 + 5x)' = 3x^2 + 5 \).
5. Product rule
When two functions are multiplied, each is differentiated in turn:
$$ \large (f(x) \cdot g(x))' = f'(x) \cdot g(x) + f(x) \cdot g'(x) $$
Example: If \( \large f(x) = x^2 \) and \( \large g(x) = \sin x \), then
$$ \large (x^2 \cdot \sin x)' = 2x \cdot \sin x + x^2 \cdot \cos x $$
6. Quotient rule
For fractions, a similar rule applies:
$$ \large \left( \frac{f(x)}{g(x)} \right)' = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{g(x)^2} $$
Example: \( \large \left( \frac{x^2}{x+1} \right)' = \frac{2x(x+1) - x^2}{(x+1)^2} = \frac{x(x+2)}{(x+1)^2} \).
7. Chain rule
When one function consists of another, the chain rule is used:
$$ \large (f(g(x)))' = f'(g(x)) \cdot g'(x) $$
Example: \( \large (\sin(3x))' = \cos(3x) \cdot 3 = 3 \cdot \cos(3x) \).
Graphical and practical consequences
The seven rules make it possible to differentiate most functions encountered in practice. They also show that the derivative is not merely an abstract tool, but a system capable of handling even complex relationships — from simple polynomials to composite exponential and trigonometric functions.
Example: combined functions
Find the derivative of \( \large f(x) = (2x^2 + 3x) \cdot e^x \).
Here both the product rule and the sum rule must be used:
$$ \large f'(x) = (4x + 3) \cdot e^x + (2x^2 + 3x) \cdot e^x = e^x \cdot (2x^2 + 7x + 3) $$
This shows that even a fairly complex function can be differentiated by combining a few simple rules.
Summary
A function is differentiable if it has a well-defined tangent at every point. Once a function is differentiable, the rules of differentiation can be used to find its derivative quickly without relying on the limit definition. These rules form the backbone of all further work in differential calculus.