Limits and Derivatives - Algebra of derivative of functions

Derivative as a Rate of Change: The derivative of a function $f(x)$ represents the instantaneous rate of change of $f(x)$ with respect to $x$. Geometrically, if you look at a curve on a graph, the derivative at any point $(x, f(x))$ is the slope of the tangent line drawn to the curve at that specific point. As the interval between two points on the curve approaches zero, the secant line transforms into this tangent line.

First Principle of Derivative: This is the fundamental definition used to calculate the derivative. For a function $f(x)$, the derivative $f'(x)$ is defined as the limit $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$. Visually, this captures the ratio of the change in the vertical 'rise' to the horizontal 'run' as the distance between points becomes infinitesimally small.

Sum and Difference Rule: The derivative of the sum or difference of two functions is simply the sum or difference of their individual derivatives. If $h(x) = f(x) \pm g(x)$, then $h'(x) = f'(x) \pm g'(x)$. On a graph, adding two functions results in a new shape whose slope at any point is the combined vertical steepness of the original two functions.

Constant Multiple Rule: If a function is multiplied by a constant $k$, its derivative is also multiplied by that same constant: $\frac{d}{dx}[k \cdot f(x)] = k \cdot f'(x)$. Visually, multiplying a function by a constant stretches or compresses the graph vertically, which scales the slope of the tangent at every point by that factor $k$.

Product Rule (Leibniz Rule): When differentiating the product of two functions, $u(x)$ and $v(x)$, the derivative is not just the product of their derivatives. Instead, it follows the pattern: the derivative of the first times the second, plus the first times the derivative of the second. This accounts for how both functions vary simultaneously.

Quotient Rule: For a function that is the ratio of two functions $\frac{u(x)}{v(x)}$ (where $v(x) \neq 0$), the derivative is calculated as 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.

Power Rule: For any function of the form $f(x) = x^n$, where $n$ is a real number, the derivative is $f'(x) = nx^{n-1}$. This rule is the workhorse of polynomial differentiation and illustrates how the degree of the function drops by one as we find its rate of change.