Differentiable

In mathematics, specifically calculus, 'differentiable' describes a function that possesses a derivative at a given point or over a specified interval. This implies the function is smooth (i.e., no sharp corners or breaks), and a tangent line can be drawn to the curve at any point where it is differentiable. A differentiable function can also be continuously differentiable, meaning its derivative is itself a continuous function. The concept of differentiability is crucial for understanding rates of change, optimization, and modeling real-world phenomena. A function must also be continuous to be differentiable, though the reverse is not always true.

Differentiable meaning with examples

• The function f(x) = x² is differentiable everywhere because its derivative, f'(x) = 2x, exists for all real numbers. This allows us to easily calculate the slope of the tangent line at any point and analyze the function's rate of change. We can use differentiation to model the motion of an object or finding the velocity.
• Although f(x) = |x| is continuous at x=0, it is not differentiable at that point because it has a sharp corner. There is no uniquely defined tangent line at the origin. The left-hand derivative and the right-hand derivative are not equal. The function is, however, differentiable everywhere else.
• In physics, position functions are often differentiable to allow for calculations of velocity and acceleration. When modeling the distance travelled by a car, a differentiable function can be used to define its path, with its first and second derivatives describing the car's velocity and acceleration, respectively. This lets you measure rates of change.
• When applying optimization techniques, functions representing cost or profit need to be differentiable to find maxima and minima. This uses concepts such as the derivative tests to find the minimum and maximum values of a function. Using these types of functions makes it easier to find maximum profits and the minimum costs.
• The solution to a differential equation is often sought within a space of differentiable functions. Mathematicians choose functions that can have multiple derivatives, allowing for a wide variety of solutions. Differentiability helps to apply various techniques like the separation of variables for solving differential equations.