The derivative represents the instantaneous rate of change of a function. While average rate of change measures the slope between two points, the derivative finds the slope at a single point by taking the limit as the interval between points approaches zero.
Formally, the derivative is defined as:
\(f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}\)
This limit definition, also known as the difference quotient, can be visualized as taking secant lines between two points and letting those points get infinitely close together until we obtain the tangent line. The slope of this tangent line is the derivative.
For example, to find the derivative of \(f(x) = x^2\) at \(x = 2\):
\(f'(2) = \lim_{h \to 0} \frac{(2 + h)^2 - 2^2}{h}\)
\(= \lim_{h \to 0} \frac{4 + 4h + h^2 - 4}{h}\)
\(= \lim_{h \to 0} \frac{4h + h^2}{h}\)
\(= \lim_{h \to 0} (4 + h) = 4\)
The power rule is one of the fundamental rules of differentiation:
For any function \(f(x) = x^n\), its derivative is: \[f'(x) = nx^{n-1}\]
Trigonometric functions have their own unique derivatives that must be memorized but can be understood through the limit definition. The most fundamental is the derivative of sine:
\(\frac{d}{dx}[\sin(x)] = \cos(x)\)
This relationship begins a pattern of trigonometric derivatives:
\(\frac{d}{dx}[\cos(x)] = -\sin(x)\)
\(\frac{d}{dx}[\tan(x)] = \sec^2(x)\)
For natural exponential and logarithmic functions:
\(\frac{d}{dx}[e^x] = e^x\)
\(\frac{d}{dx}[\ln(x)] = \frac{1}{x}\)
The natural exponential function \(e^x\) is unique because it is its own derivative. This special property makes it fundamental in calculus and explains its importance in growth and decay models.
When differentiating the product of two functions, we cannot simply multiply their derivatives. Instead, we use the product rule, which states that the derivative of a product is the first function times the derivative of the second, plus the second function times the derivative of the first.
\(\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)\)
Similarly, for quotients, we use the quotient rule. The derivative of a quotient is the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator.
\(\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{[g(x)]^2}\)
A helpful mnemonic for the quotient rule is "low d-high minus high d-low, over low low" where "low" refers to the denominator and "high" to the numerator.
The chain rule is essential for differentiating composite functions. It states that when differentiating a composite function, we multiply the derivative of the outer function (evaluated at the inner function) by the derivative of the inner function.
If \(h(x) = f(g(x))\), then \(h'(x) = f'(g(x)) \cdot g'(x)\)
For example, to differentiate \(h(x) = \sin(x^2)\), we identify \(\sin(x)\) as the outer function and \(x^2\) as the inner function. The derivative would be:
\(h'(x) = \cos(x^2) \cdot 2x\)
Test your understanding of differentiation concepts.