Derivative - Differentiation Rules and Advanced Concepts

Rules of Differentiation Introduction Differentiation is the process of finding the derivative of a function, which represents the rate of change at any given point. Rather than calculating derivatives from first principles every time, mathematicians have developed a set of systematic rules that allow us to quickly find derivatives of even complex functions. These rules form the foundation of computational calculus and are essential tools for solving optimization problems, analyzing motion, and much more. In this section, we'll learn the rules for differentiating basic functions, combining functions together, understanding when functions fail to be differentiable, and computing higher-order derivatives. Rules for Basic Functions The Constant Rule The simplest differentiation rule concerns functions that don't change at all. If $f(x) = c$ where $c$ is any constant, then the derivative is: $$f'(x) = 0$$ Why? The derivative measures how much a function changes. A constant function never changes, so its rate of change is zero everywhere. The Power Rule For functions of the form $f(x) = x^n$ where $n$ is any real number, the derivative is: $$f'(x) = nx^{n-1}$$ Example: If $f(x) = x^3$, then $f'(x) = 3x^2$. If $g(x) = x^{-2}$, then $g'(x) = -2x^{-3}$. This rule works for all real exponents, including fractional and negative ones. It's one of the most frequently used rules in calculus. The Exponential Rule The natural exponential function $e^x$ has the remarkable property that its derivative is itself: $$\frac{d}{dx}(e^x) = e^x$$ This property makes the exponential function fundamental in modeling continuous growth and decay. The Logarithmic Rule For the natural logarithm function (logarithm with base $e$), the derivative is: $$\frac{d}{dx}(\ln(x)) = \frac{1}{x}$$ Note that this is only defined for $x > 0$, since the natural logarithm itself is only defined for positive values. The Trigonometric Rules The standard trigonometric functions have well-established derivatives: $$\frac{d}{dx}(\sin(x)) = \cos(x)$$ $$\frac{d}{dx}(\cos(x)) = -\sin(x)$$ $$\frac{d}{dx}(\tan(x)) = \sec^2(x)$$ Tip: Notice the pattern with sine and cosine—they cycle through derivatives of each other (with sign changes). Tangent's derivative involves the secant function, which is $\sec(x) = 1/\cos(x)$. Rules for Combining Functions In practice, most functions are combinations of simpler functions. The rules below show how to find derivatives of sums, products, quotients, and compositions. The Sum Rule If $f(x) = u(x) + v(x)$, then: $$f'(x) = u'(x) + v'(x)$$ The derivative of a sum is simply the sum of the derivatives. This makes it easy to break complex functions into pieces. Example: If $f(x) = x^2 + \sin(x)$, then $f'(x) = 2x + \cos(x)$. The Difference Rule Similarly, if $f(x) = u(x) - v(x)$, then: $$f'(x) = u'(x) - v'(x)$$ The derivative of a difference is the difference of the derivatives. The Product Rule When you need to differentiate the product of two functions, you cannot simply multiply the derivatives. Instead, if $f(x) = u(x) \cdot v(x)$: $$f'(x) = u'(x)v(x) + u(x)v'(x)$$ A common way to remember this: "first derivative times second, plus first times second derivative." Example: If $f(x) = x^2 \sin(x)$, then $f'(x) = 2x \sin(x) + x^2 \cos(x)$. The Quotient Rule For quotients, if $f(x) = \frac{u(x)}{v(x)}$ (where $v(x) \neq 0$): $$f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{[v(x)]^2}$$ This is often remembered as: "low derivative high, minus high derivative low, over low squared." The order matters here! It's the derivative of the numerator times the denominator, minus the numerator times the derivative of the denominator. Example: If $f(x) = \frac{x^2}{e^x}$, then $f'(x) = \frac{2x \cdot e^x - x^2 \cdot e^x}{(e^x)^2} = \frac{e^x(2x - x^2)}{e^{2x}} = \frac{2x - x^2}{e^x}$. The Chain Rule The chain rule applies when you have a composition of functions—one function nested inside another. If $y = f(g(x))$, then: $$\frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx}$$ where $u = g(x)$. Alternatively written: If $y = f(g(x))$, then $y' = f'(g(x)) \cdot g'(x)$. Motivation: The chain rule lets you "peel back" the layers of composition. The inner function's rate of change is multiplied by the outer function's rate of change. Example: If $f(x) = \sin(x^2)$, we can write this as $f(x) = \sin(u)$ where $u = x^2$. Then: $\frac{df}{du} = \cos(u) = \cos(x^2)$ $\frac{du}{dx} = 2x$ $\frac{df}{dx} = \cos(x^2) \cdot 2x = 2x\cos(x^2)$ Continuity and Differentiability The Relationship Between Continuity and Differentiability An important fact: If a function is differentiable at a point, then it must be continuous at that point. However, the converse is not true—a function can be continuous at a point without being differentiable there. Example: The absolute value function $f(x) = |x|$ is continuous everywhere, including at $x = 0$. However, it is not differentiable at $x = 0$ because the function has a sharp corner (a "kink") at that point. Types of Points Where Differentiability Fails Even if a function is continuous, it can fail to be differentiable in several ways: Jump Discontinuity: If a function has a jump discontinuity (a sudden break where the left and right limits don't match), then it cannot be differentiable there. The function's value "jumps," so no tangent line exists. Kink or Cusp: A kink is a sharp corner where the tangent line changes direction abruptly. At such a point, the left-hand slope (approaching from the left) and right-hand slope (approaching from the right) are different. This means the derivative doesn't exist. The absolute value function $|x|$ at $x = 0$ is a classic example. Vertical Tangent: Sometimes a function is smooth and continuous, but the tangent line becomes vertical. This means the slope becomes infinitely steep, and the derivative is undefined as a finite number. For example, $f(x) = \sqrt[3]{x}$ (the cube root) has a vertical tangent at $x = 0$. Functions That Are Differentiable Everywhere Polynomial functions are differentiable at every real number. This is because polynomials are built from powers of $x$ using addition, and the power rule combined with the sum rule guarantees a well-defined derivative everywhere. Higher-Order Derivatives What Are Higher-Order Derivatives? A derivative measures the rate of change of a function. But the derivative itself is a function, so you can take its derivative! This leads to higher-order derivatives. The first derivative $f'(x)$ measures the rate of change of $f(x)$. The second derivative $f''(x)$ (or $\frac{d^2f}{dx^2}$) is the derivative of $f'(x)$. It measures the rate of change of the rate of change—in other words, the acceleration or curvature. The third derivative $f'''(x)$ (or $f^{(3)}(x)$) is the derivative of the second derivative. In general, the $n$-th derivative $f^{(n)}(x)$ is obtained by differentiating $n$ times. Example: If $f(x) = x^4$, then: $f'(x) = 4x^3$ $f''(x) = 12x^2$ $f'''(x) = 24x$ $f^{(4)}(x) = 24$ $f^{(5)}(x) = 0$ (and all further derivatives are 0) Differentiability Classes The notation $C^n$ is used to describe the smoothness of a function: A function belongs to class $C^0$ if it is continuous. A function belongs to class $C^1$ if its first derivative exists and is continuous. A function belongs to class $C^2$ if its second derivative exists and is continuous. In general, a function belongs to class $C^n$ if its $n$-th derivative exists and is continuous. This matters because continuous derivatives ensure smooth behavior. A function in $C^1$ has no kinks or corners. Smooth and Infinitely Differentiable Functions A function is called smooth (or infinitely differentiable) if it has derivatives of all orders. The notation for this is $C^\infty$. Examples of smooth functions: $e^x$ (you can differentiate it infinitely many times and always get $e^x$ again) $\sin(x)$ and $\cos(x)$ (their derivatives cycle through each other indefinitely) Polynomial functions (eventually the derivatives become 0, but that counts as existing) Not all functions are smooth. For instance, $|x|$ is not even $C^1$ because its first derivative has a jump at $x = 0$.