NUMERICAL METHODS AND ANALYSIS NMA 312 Prof. Samuel Okolie, Prof. Yinka Adekunle & Dr. Seun Ebiesuwa

ANALYTICAL DIFFERENTIATION DERIVATION OF ALGEBRAIC FUNCTIONS (Polynomials, Trignometrical, Logarithmic, Exponential and other functions in physics and other sciences)

1. A function is a relationship between two or more variables. eg. y = x2 + 3x + 2; implies y = f(x) 2. Definition of the Derivative of a function: Let y = f(x) define a function f. If the limit dy/dx = Lim (Δy/Δx) Δx 0 meaning f(x) = Lim f(x + Δx) – f(x) exists Δx 0 Δx and is finite, we call this limit the derivative of y wrt x and say that f is differentiable at x

3. The derivative of a constant is zero. d/dx (c) = 0 where c is a constant 4. the derivative wrt x of cxn is cnxn-1 when n is any positive integer, and c is a constant i.e if y = cx n dy/dx = c n x n-1 [proof available in calculus textbooks]

5. If y = u + v, u and v are functions of x then dy/dx = du/ dx + dv/dx 6. If y = (uv) and u and v are functions of x then dy/dx = u(x) dv/dx + du/dx (v(x)) product rule Generally extending the above to more than one product we have y = u v w, where u,v,w are functions of x dy/dx = uv dw/dx + uw dv/dx + vw du/dx

7. If y = u/v where v ≠ o dy/dx = d/dx (u/v) = v du/dx – udv/dx quotient rule V2 8. If u = g(x) is a differentiable function of x and n is a positive integer or o, then d/dx(U)n = n Un-1 du/dx if y = f(g) dy/dx = d f(g) . d(g) Chain rule d(g) dx 9. If y = e ax where e is Naparian constant. dy/dx = a eax a is constant Exponential Rule

10. If y = log x dy/dx = I/x Logarithmic Rule Similarly if y = log(ax) where a is constant dy/dx = a/x 11. If y = bx dy/dx = bxlog b y = sin ax a is constant dy/dx = a cos ax

y dy/dx cos ax - a sin ax sec ax sec ax tan ax tan ax a sec2 ax Similarly we have the following trigonometrical, differentiation. y dy/dx cos ax - a sin ax sec ax sec ax tan ax tan ax a sec2 ax cosec ax - a cosec ax cof ax cot ax - a cosec2 ax

13. (Arch-functions) If y = sin -1 x/a dy/dx = 1 √(a2-x2) Similarly the differential of the arch functions are: y dy/dx cos-1 x/a 1 √(a2-x2) tan-1 x/a a/ (a2 + x2) cot-1 x/a -a/(a2 + x2) sec-1 x/a a lxl √(a2-x2) cosec-1 x/a -a lxl √(x2-a2)

14. Implicit Differentiation In an equation where the dependent and independent variables are mixed up: we have implicit differentiation. If x3 – 3xy + y2 = 2, find dy/dx. 3x2 – (3x dy/dx +3y) + 2y dy/dx = 0 dy/dx (2y – 3x) = - 3x2 + 3y dy/dx = 3y-3x2 2y – 3x

Exercise: 1. Prove d (uv)/d2x = d2 uv/d2x + 2 du/dx. dv/dx + ud2v/dx 2 Exercise: 1. Prove d (uv)/d2x = d2 uv/d2x + 2 du/dx . dv/dx + ud2v/dx 2. If y = ax, show that dy/dx = ax log a 3. y = x , find dy/dx √(x2- 4) 4. x2y + x y2 = 6 (x2 + y2); find dy/dx. 5. y = √(2t + t2), t = 3x + 3, find dy/dx 6. xy + 2x + 3y = 1. Find dy/dx 7. y = log (ax + a-x), find dy/dx. 8. y = √(1-x )/(1 + x2 ), find dy/dx