Differential Calculus
Master the techniques of finding derivatives for various functions.
The Core of Differentiation
While the "first principles" definition of a derivative is foundational, it's often cumbersome for complex functions. Differential calculus provides a set of rules and formulas that streamline the process of finding derivatives.
These rules are derived from the limit definition but allow us to compute derivatives much more efficiently. Mastering these rules is the key to unlocking the power of calculus for analyzing functions.
Derivatives from First Principles
Every differentiation rule is derived from the limit definition of a derivative, also known as finding the derivative from "first principles".
It calculates the slope of the secant line between two points on the curve, and then takes the limit as the distance between these points approaches zero. This turns the secant line into the tangent line at a single point.
f'(x) = lim (h→0) [f(x + h) - f(x)] / hIn this formula:
his a very small change in the x-value.f(x + h) - f(x)is the corresponding change in the y-value.- The entire fraction,
[f(x + h) - f(x)] / h, represents the slope of the secant line through the points(x, f(x))and(x+h, f(x+h)).
The visualizer below shows how the secant line (blue) approaches the tangent line (red) as the distance `h` gets smaller. The derivative, f'(x), is the slope of this red tangent line, which is what the slope of the secant line approaches as h heads to zero.
Example problems on Differentiation from First Principles
Basic Differentiation Rules
| Rule Name | Formula |
|---|---|
| Constant Rule | d/dx (c) = 0 |
| Power Rule | d/dx (xⁿ) = nxⁿ⁻¹ |
| Constant Multiple Rule | d/dx [cf(x)] = c · f'(x) |
| Sum/Difference Rule | d/dx [f(x) ± g(x)] = f'(x) ± g'(x) |
Product & Quotient Rules
| Rule Name | Formula |
|---|---|
| Product Rule | d/dx [f(x)g(x)] = f(x)g'(x) + g(x)f'(x) |
| Quotient Rule | d/dx [f(x)/g(x)] = [g(x)f'(x) - f(x)g'(x)] / [g(x)]² |
Trigonometric Derivatives
| Rule Name | Formula |
|---|---|
| Sine | d/dx (sin x) = cos x |
| Cosine | d/dx (cos x) = -sin x |
| Tangent | d/dx (tan x) = sec² x |
Implicit Differentiation
Sometimes, functions are defined implicitly, meaning `y` is not given as an explicit function of `x`. For example, the equation of a circle `x² + y² = 25` defines `y` implicitly.
Implicit differentiation allows us to find the derivative `dy/dx` without having to solve for `y` first.
Example: Find dy/dx for x² + y² = 25
- Differentiate both sides with respect to x:
d/dx (x²) + d/dx (y²) = d/dx (25) - Apply differentiation rules. Remember to use the Chain Rule for y:
The derivative of `x²` is `2x`.
The derivative of `y²` is `2y · dy/dx` (by the Chain Rule, since y is a function of x).
The derivative of `25` is `0`.
This gives us:2x + 2y · dy/dx = 0 - Solve for dy/dx:
2y · dy/dx = -2xdy/dx = -2x / 2y = -x / y
This result tells us that the slope of the tangent line at any point (x, y) on the circle is `-x/y`.
The Chain Rule
One of the most powerful tools in differentiation is the Chain Rule. It is used to find the derivative of composite functions (a function inside another function).
If h(x) = f(g(x)), then h'(x) = f'(g(x)) · g'(x)In words: "the derivative of the outer function (with the inner function left alone) times the derivative of the inner function."
Example: Find the derivative of sin(x²)
- Outer function f(u) = sin(u), so f'(u) = cos(u)
- Inner function g(x) = x², so g'(x) = 2x
- Applying the rule:
f'(g(x)) · g'(x) = cos(x²) · 2x = 2x cos(x²)
Calculation of Limits using L'Hospital's Rule
L'Hospital's Rule is a powerful method used to evaluate limits of indeterminate forms like 0/0 or ∞/∞.
If you have a limit lim (x→c) [f(x) / g(x)] which results in 0/0 or ∞/∞, L'Hospital's Rule states that this limit is equal to lim (x→c) [f'(x) / g'(x)], provided the limit on the right side exists.
Example (0/0 form): Find lim (x→0) sin(x)/x
- Check the form:
Substituting x=0 gives sin(0)/0 = 0/0. This is an indeterminate form. - Apply L'Hospital's Rule:
Differentiate the numerator and the denominator: d/dx(sin(x)) = cos(x) and d/dx(x) = 1. - Evaluate the new limit:
The limit becomeslim (x→0) cos(x)/1. Substituting x=0 gives cos(0)/1 = 1.
Example (∞/∞ form): Find lim (x→∞) eˣ/x²
- Check the form:
As x→∞, both eˣ and x² approach ∞. This is an ∞/∞ form. - Apply L'Hospital's Rule:
Differentiate top and bottom:lim (x→∞) eˣ/(2x). This is still an ∞/∞ form. - Apply L'Hospital's Rule again:
Differentiate again:lim (x→∞) eˣ/2. - Evaluate the final limit:
As x→∞, eˣ/2 approaches ∞. The limit is ∞.