Applications of Derivatives explore how rates of change and slopes of tangent lines can be used to analyze and solve real-world problems. This chapter builds on the concepts of differentiation to provide practical tools for various applications.
If y = f(x) , then rate of change of y w.r.t x is : dy/dx if x = f(t) , y = f(t) , then by chain rule: dy/dx = (dy/dt)/(dx/dt) ⇒ dy/dx = (dy/dt) • (dt/dx)
Increasing or decreasing function
Increasing Function :- f(x) defined on ]a,b[ : x1 < x2 ⇒ f(x1) < f(x2) for x1,x2 ∈ ]a,b[ x1 > x2 ⇒ f(x1) > f(x2) for x1,x2 ∈ ]a,b[ Decreasing Function :- f(x) defined on ]a,b [ : x1 < x2 ⇒ f(x1) > f(x2) for x1,x2 ∈ ]a,b[ x1 > x2 ⇒ f(x1) < f(x2) for x1,x2 ∈ ]a,b[
Tangent and Normals
Slope of tangent: m = dy/dx Equation of tangent[at point(x1, y1)]: y - y1 = m(x - x1) Equation of normal: y - y1 = -1/m(x - x1)
Maxima & minima
First Derivative test: f'(x) = 0 • solve f'(x) = 0 → critical points Second derivative test: f”(x) > 0 , f"(x) < 0 • if f"(x) > 0 → Minimum • If f"(x) < 0 → Maximum
Approximation (Linear Approximation)
f(x + △x) ≈ f(x) + f'(x) · △x • Small change formula: dy = f'(x)dx