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

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

3. 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 Δx
and is finite, we call this limit the derivative of y wrt x and say that f is differentiable at x

4. 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 [proof available in calculus textbooks]

5. 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

6. 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

7. 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

8. 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

9. 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)

10. 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

11. 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