Calculus - Derivatives

Definition

The derivative of a function represents the instantaneous rate of change of the function value with respect to its variable. Geometrically, it is the slope of the tangent line to the graph of the function at a given point. $f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} = \lim_{z \to x} \frac{f(z) - f(x)}{z - x}$

Interpretation

• Rate of change: f’(x) = instantaneous rate of change
• Slope: f’(x) = slope of tangent line to f at x
• Velocity: if s = position, then ds/dt = velocity
• Acceleration: if v = velocity, then dv/dt = acceleration

Basic Derivative Rules

Power Rule: $\frac{d}{dx}(x^n) = nx^{n-1}$

Constant Rule: $\frac{d}{dx}(c) = 0$

Constant Multiple: $\frac{d}{dx}[cf(x)] = cf'(x)$

Sum/Difference: $\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)$

Product Rule: $\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$

Quotient Rule: $\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)g(x) - f(x)g'(x)}{[g(x)]^2}$

Chain Rule: $\frac{d}{dx}[f(g(x))] = f'(g(x))g'(x)$

Inverse Function: $(f^{-1})'(y) = \frac{1}{f'(f^{-1}(y))}$

Derivatives of Elementary Functions

Trigonometric:

• d/dx(sin x) = cos x
• d/dx(cos x) = -sin x
• d/dx(tan x) = sec² x
• d/dx(cot x) = -csc² x
• d/dx(sec x) = sec x tan x
• d/dx(csc x) = -csc x cot x

Inverse Trigonometric:

• d/dx(arcsin x) = 1/√(1-x²)
• d/dx(arccos x) = -1/√(1-x²)
• d/dx(arctan x) = 1/(1+x²)
• d/dx(arcsec x) = 1/(|x|√(x²-1))
• d/dx(arccsc x) = -1/(|x|√(x²-1))
• d/dx(arccot x) = -1/(1+x²)

Exponential and Logarithmic:

• d/dx(e^x) = e^x
• d/dx(a^x) = a^x ln a
• d/dx(ln x) = 1/x
• d/dx(log_a x) = 1/(x ln a)

Hyperbolic:

• d/dx(sinh x) = cosh x
• d/dx(cosh x) = sinh x
• d/dx(tanh x) = sech² x

Implicit Differentiation

Differentiate both sides of equation with respect to x, treating y as function of x.

Example: x² + y² = r² 2x + 2y(dy/dx) = 0 → dy/dx = -x/y

Logarithmic Differentiation

Take ln of both sides to simplify complicated functions.

Applications

• Used to find maxima and minima (optimization)
• Basis for Differential Equations
• Velocity and acceleration in physics

Relationships

• Related: Fundamental Theorem of Calculus, Integral Calculus