Differentiation Lecture Note

Differentiation 1 Differentiation 1.1 Definition of the Derivative The derivative of a function measures the rate at which the function’s value changes with respect to changes in its input. It is formally defined as: f (x + h) − f (x) , h→0 h f ′ (x) = lim provided the limit exists. 1.1.1 Geometric Interpretation The derivative f ′ (x) represents the slope of the tangent line to the curve y = f (x) at a point x. 1.2 Basic Rules of Differentiation Let u(x) and v(x) be differentiable functions, and c be a constant. Then: d [c] dx 1. Constant Rule: 2. Power Rule: d [xn ] dx = 0. = nxn−1 for any real number n. 3. Constant Multiple Rule: 4. Sum Rule: d [u(x) dx 5. Difference Rule: 6. Product Rule: 7. Quotient Rule: d [cu(x)] dx + v(x)] = d [u(x) dx h u(x) v(x) i + dv . dx − v(x)] = d [u(x)v(x)] dx d dx du dx = = c du . dx du dx − dv . dx = u′ (x)v(x) + u(x)v ′ (x). u′ (x)v(x)−u(x)v ′ (x) , [v(x)]2 1 provided v(x) ̸= 0. 1.3 Chain Rule If y = f (g(x)), then: dy = f ′ (g(x)) · g ′ (x). dx The chain rule is essential for differentiating composite functions. 1.4 Higher-Order Derivatives The n-th derivative of a function is denoted by f (n) (x). For example: f ′′ (x) = 2 2.1 2.1.1 d3 d2 (3) [f (x)] and f (x) = [f (x)]. dx2 dx3 Examples of Differentiation Basic Examples Example 1: Power Rule Differentiate f (x) = x5 : f ′ (x) = 5x5−1 = 5x4 . 2.1.2 Example 2: Product Rule Differentiate f (x) = (x2 + 1)(x3 − x): f ′ (x) = (x2 + 1) d 3 d (x − x) + (x3 − x) (x2 + 1). dx dx f ′ (x) = (x2 + 1)(3x2 − 1) + (x3 − x)(2x). 2.1.3 Example 3: Quotient Rule Differentiate f (x) = x2 +1 : x3 −x f ′ (x) = (2x)(x3 − x) − (x2 + 1)(3x2 − 1) . (x3 − x)2 2 2.2 2.2.1 Chain Rule Example Example 4: Composite Function Differentiate f (x) = sin(x2 ): f ′ (x) = cos(x2 ) · 2x. 3 3.1 Applications of Differentiation Finding Tangent Lines The equation of the tangent line to f (x) at x = a is given by: y − f (a) = f ′ (a)(x − a). 3.1.1 Example 5: Tangent Line Find the equation of the tangent line to f (x) = x2 + 2x + 1 at x = 1: f ′ (x) = 2x + 2, f ′ (1) = 4. f (1) = 12 + 2(1) + 1 = 4. The tangent line is: y − 4 = 4(x − 1) or y = 4x. 3.2 Optimization Optimization involves finding the maximum or minimum values of a function. Critical points occur where f ′ (x) = 0 or f ′ (x) is undefined. 3.2.1 Example 6: Optimization Problem Find the maximum of f (x) = −x2 + 4x + 5: f ′ (x) = −2x + 4. Setting f ′ (x) = 0: −2x + 4 = 0 =⇒ x = 2. 3 To classify, use the second derivative: f ′′ (x) = −2. Since f ′′ (x) < 0, x = 2 is a local maximum. The maximum value is: f (2) = −(2)2 + 4(2) + 5 = 9. 3.3 Related Rates Related rates involve finding how one quantity changes with respect to another using derivatives. 3.3.1 Example 7: Related Rates A spherical balloon is being inflated. The radius r is increasing at a rate of 2 cm/s. Find the rate of change of the volume when r = 5 cm. 4 Volume of a sphere: V = πr3 . 3 Differentiate with respect to t: dV dr = 4πr2 . dt dt Substitute r = 5 and dr dt = 2: dV = 4π(5)2 (2) = 200π cm3 /s. dt 4 Conclusion Differentiation is a powerful tool for analyzing the behavior of functions. It allows us to compute rates of change, find slopes of tangent lines, and solve real-world problems involving optimization and related rates. Mastery of differentiation rules and their applications is crucial for advanced topics in calculus. 4