7. Formal differentiation rules
Derivatives of elementary functions
Elem. function Its derivative Composed with inner function Composite function f (x) ( f (x)) f (x) ( f (g(x)) f (g(x)) g(x) example
2 1 Power function (1 x ) 2 1 1 (x ) x ((g(x)) ) (g(x)) g(x) 2 1 x 1 (1 x ) 2 (2x) (α: real constant) 2
Root function 1 x Use x x to reduce roots to power functions (β ≠ 0 is real)
Exponential x x g(x) g(x) tg x tg x 1 (a ) a ln a (a ) (a ln a) g(x) 2 2 ln 2 2 function a x cos x
(0 < a ≠ 1 real x x g(x) g(x) (e ) e (e ) e g(x) x x x constant) e e (1) e
Logarithm 1 function g (x) (loga x) , (loga g(x)) , g(x) 0 sin x loga x x ln a g(x)ln a (lg(cos x)) cos x ln10 (0 < a ≠ 1 real if x 0 logarithmic derivative of g(x): constant)
1 g(x) x x e ln x loge x (ln x) , if x 0 (ln g(x)) , ha g(x) 0 (ln (1 e )) x g(x) 1 e x
1 sin x (sin x) cos x (sin g(x)) cos g(x) g(x) (sin(ln x)) cos(ln x)) x
(cos (x2 1)) cos x (cos x) sin x (cos g(x)) sin g(x) g(x) sin(x2 1)) 2x
1 (tgx) g(x) tg2 x 1 t an x cos2 x (tgg(x)) (tg(tgx)) cos2 (g(x)) cos2 (tgx) tg2 x 1 1 1 (ctg x) g(x) ctgx ctg x sin2 x (ctg g(x)) sin2 x sin 2 (g(x)) ctg2 x 1
General rules
1. Constant factor rule: (cf (x)) cf (x)
2. Sum rule: ( f (x) g(x)) f (x) g(x)
3. Difference rule: ( f (x) g(x)) f (x) g(x)
4. Product rule: ( f (x) g(x)) f (x)g(x) f (x)g(x)
f (x) f (x)g(x) f (x)g(x) 5. Quotient rule: 2 g(x) (g(x))
6. Chain rule (composite function): ( f (g(x)) f (g(x)) g(x)
7. Logarithmic differentiation: f (x) f (x)(ln f (x)) Using logarithm we can reduce product to sum, quotient to difference, power to product, e. g.:: (x 3 1)sin 2 x a) f (x) f (x) f (x) ln(x 3 1) ln(sin 2 x) ln(e x e 2 ) e x e 2 (x 3 1)sin 2 x 3x 2 2sin x cos x e x x 2 3 2 x 2 e e x 1 sin x e e 1 1 x x x 1 2 1 1 b) f (x) x x f (x) f (x) ln x f (x) (x ln x) f (x) (x ln x x x ) 1 ln x x x x 2
PROBLEM 1 Differentiate these functions using rules for fundamental operations: x 2 x 3 a) 1 x b) 3sin x tgx 2 2 c) 2x ( 2x 33 x) 2 6
x 2005 2 2 2 e x d) 3tgx 3 lg x e) sin x cos x f) x 2006 x
4 2 e sin x cos x x x 2 x ctg x g) h) 5 3 i) 2 sin x cos x 2x x 9x x ln x x(e x 1) j) log0,5 x PROBLEM 2 Differentiate these functions using the chain rule for composite functions:
a) log (x) b) e2x c) cos x d) 4x 2 2
x e 2,71828 cosxsinx 2005 2006 e) (e x ) f) e g) sin 2x cos 2x h) x x i) cos(cos x) j) exp ( 1 x2 ) k) exp (ex ) l) 2 1 x x 100 1 x m) 1 1 x n) ln o) log2 p) 1 1 x 2 100 q) sin(cos ( x )) r) tg(ctg(x ))
PROBLEM 3 Differentiate these functions using the appropriate rules:
e2x e2x a) 1 e4x 1 43x1 b) x3 (tg3x ctg2x) c) ln 2 3 e2x e2x
3 2 x 3 x x x 1 2 2 d) 3 log3 (1 x ) e) 2 f) (1 x ) 1 x 2x 2 sin 2x g) h) ex sin2x i) xe ln (1 ex ) e x2 cos 2 x
4 x 4 x 4 cos2 x cos x2 j) (2ctgx lg x)2 k) l) tg4x x2
3x tgx m) esinx ln (sin x) n) ctg(tgx) tg(ctg x) o*) 3 ln 2x
(1 2x 2 )sin x p*) q*) (cos x)tg x r*) x x x 2 e2x x 2 1
sinxcosx x s*) sin x cos x t*) x
Domain of definition, differentiability, derivative
PROBLEM 4 What is the domain of definition and where is it differentiable? Determine the derivative! (also plot the graph of the original function and give its range.)
3 2 2ln(1x) a+) x 1 b) log2 (1 cos x) c+) e d) ctg x 1 x 1 e) sin x f+) cos x 1 1 g) h) 2 2 x x 3x 2 x 1 i) 2x x2 j) ln (tgx) k+) l+) ln x x 1 x
Equation of tangent line
PROBLEM 5 Detemine the tangent line of f (x) at x0! (+-problem: also plot the graph of f (x) and its
tangent line.) Approximate the value of f (x) using the tangent line equation at x=x0+0.1
or at x=x0–0.1. x2 a+) f (x) x 1, x 5 b+) f (x) 3x 4, x 1 0 2 0 2x 5 c+) f (x) , x 0 d+) f (x) 1 cos 2x, x x 2 0 0 4
2 x1 e) f (x) ln (x 2x 2), x0 1 f) f (x) 2xe 1, x0 1