Tiêu đề MATHEMATICS FOR PHYSICS - 2.pdf ✅

CLASS VIII PHYSICS DIFFERENTIATION Constant quantity: If the value of a quantity remains the same in a mathematical operation, it is called a con- stant quantity. Examples: Integers (4,7,13, ... ), Fraction (1/2,4/5), π, e, etc., Variable quantity: If the quantity takes different values in a mathematical operation, it is called a variable quantity. Examples: (i) In the equation y = 2x + 3, x and y are variables. (ii) In the equation F = ma where F, m and a are variables. Variable quantities are divided into two types (a) Independent quantity. (b) Dependent quantity. In the equation y = 2x + 3, x is the independent quantity and y is the dependent quantity. Function: If corresponding to any given value of x, there exists a single definite value of y, then y is called a function of x. This is represented as y = f(x). This equation means that corresponding to one value of x, there is a single definite value of the variable y. Example : (i) Let y = 2x, when x = 1 then y = 2; when x = 2 then y = 4 ... ... Thus corresponding to each value of x, there is a definite value of y. Difference and Differential coefficient: Let y be a function of x. This is denoted as y = f(x). Here x and y are variables. Let the value of x change to x + Δx and correspondingly the value of y changes to y + Δy The change between the initial and final values of a variable quantity is called its difference. Difference in x = (x + Δx) − x = Δx; Difference in y = (y + Δy) − y = Δy The ratio of Δy/Δx is called the quotient of the two increments. When difference in x (i.e., Δx ) is very very very small i.e., almost approaching to zero, then we write Δy Δx is equal to dy dx . In mathematical language we represent the above statement as : LimitΔx→0 Δy Δx = dy dx or LtΔx→0 Δy Δx = dy dx. In this equation Δy Δx is the ratio of a small quantity Δy to another small quantity Δx. But dy/dx is a single quantity and is called the differential coefficient of y with respect to x. dy dx = LtΔx→0 Δy Δx = LtΔx→0 [ f(x+Δx)−f(x) Δx ] The rate of change of a dependent variable with respect to the independent variable is called the differential co- efficient or derivative. d dx does not mean that d is divided by dx. It is a single operator called the differential operator. Differentiation: The process of finding the differential coefficient of a function is called differentiation. Basic Theorems on Differentiation i) The derivative of a constant is zero. Let y = f(x) = c. Where c is constant. Then dy dx = d dx (c) = 0 Example: Differentiate y = 2a where a is a constant dy dx = d dx (2a) = 0 ii) The differential coefficient of x n is obtained by decreasing the power of x by unity and multiplying by n. MATHEMATICS FOR PHYSICS SYNOPSIS - 1
CLASS VIII PHYSICS If y = x n then dy dx = nxn−1 , where n may be positive or negative. Example: Differentiate y = 8x 8 w.r.t. x. Solution: dy dx = d dx (8x 8 ) = 8 d dx (x 8 ) = 8 × 8 × x 8−1 = 64x 7 iv) The derivative of the algebraic sum of two functions is equal to the algebraic sum of the derivatives of the two functions. Let y = u ± v ± w ± ⋯ ... where u, v, w.....are all functions of x. Then dy dx = d dx (u ± v ± w ± ⋯ ) = d dx (u) ± d dx (v) ± d dx (w) ± ⋯. Example: Differentiate y = 3x 4 + 2x 2 − 10x w.r.t. x. Solution: dy dx = d dx (3x 4 + 2x 2 − 10x) = d dx (3x 4 ) + d dx (2x 2 ) − d dx (10x) = 4 ⋅ 3 ⋅ x 4−1 + 2 ⋅ 2 ⋅ x 2−1 − 10 ⋅ x 1−1 = 12x 3 + 4x − 10 Rules of differentiation Product Rule: The differential coefficient of the products of two functions = 1 st function × differential coefficient of the 2 nd function +2 nd function × differential coefficient of the 1 st function. Let y = uv where u and v are functions of x. Then dy dx = d dx (uv) = u dv dx + v du dx Example: Differentiate y = x(x 2 − 2x) w.r.t. x. Solution: Here u = x, v = x 2 − 2x $ dy dx = d dx [x(x 2 − 2x)] = x d dx (x 2 − 2x) + (x 2 − 2x) d dx (x). = x(2x − 2) + (x 2 − 2x) × 1 = 3x 2 − 4x Quotient Rule: The differential coefficient of quotient of two functions = [2 nd function × derivative of the 1 st function −1 st function × derivative of 2 nd function] divided by the square of the second function. Let y = u/v where u and v are two functions of x. Then, dy dx = d dx (u/v) = v(du/dx)−u(dv/dx) v 2 Example: Differentiate y = (x 2 + 1)/(x − 1) w.r.t. x. Solution: Here u = x 2 + 1, v = x − 1. dy dx = d dx [(x 2 + 1)/(x − 1)] = (x − 1) d dx (x 2 + 1) − (x 2 + 1) d dx (x − 1) (x − 1) 2 = (x − 1) × 2x − (x 2 + 1) × 1 (x − 1) 2 = x 2 − 2x − 1 (x − 1) 2 Chain rule or function of a function rule: Let y = f(u) where y is a function of u and u is a function of x Then,dy dx = dy du ⋅ du dx Example: Differentiate y = (4x 2 − 5x + 10) 10 w.r.t. x. Solution: Let 4x 2 − 5x + 10 = u. Then y = u 10 dy du = d du (u 10) = 10u 9 ... ... − ⋯ − ⋯ du dx = d dx (4x 2 − 5x + 10) = (8x − 5) ... − ⋯ Now, dy dx = dy du ⋅ du dx = 10u 9 × (8x − 5) = 10(4x 2 − 5x + 10) 9 × (8x − 5) Now, dy dx = dy du ⋅ du dx = 10u 9 × (8x − 5) = 10(4x 2 − 5x + 10) 9 × (8x − 5)