C2 Methods of Differentiation. Definition. Let y = f(x) be a function. The derivative of f is the function whose value at x is the limit. provided this limit exists. Recall. Section 1. Fundamental Formulas for Differentiation. Formula 1.1 The derivative of a constant is 0.
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C2 Methods of Differentiation
The derivative of a constant is 0.
The derivative of the identity function f(x)=x is the constant function f'(x)=1.
If f and g are differentiable functions, then
(fg)'(x) = f(x) g'(x) + g(x) f'(x)
= u2×…×un×u1’+ u1u3×…×un×u2’+ u1u2u4×…×un×u3’
(cu)’ = cu’
Let F be the composition of two differentiable functions f and g;
F(x) = f(g(x)).
Then F is differentiable and
F'(x) = f'(g(x)) g'(x)
if f(x)=xn where n is a positive integer, then
By Chain Rule and Formula 4.1
Differentiation of Logarithmic and Exponential Functions
Read Examples 4.2- 4.4
Section 5Differentiation of
Proof of Formula
Graphs of trigonometric functions
y=cotx and y=arccotxy=secx and y=arcsecxy=cscx and y=arccscx
Section 7Differentiation of Inverse of Trigonometric Function
Proof of Formula
Section 10 Indeterminate Forms and L’Hospital Rule
(i) Evaluate limx→a f(x)/g(x) where f(a)=g(a)=0.
1.Evaluate limx→o sin3x/sin2x.
= limx→o 3cos3x/2cos2x
2.limx→o (x-sinx)/x3=limx→o (1-cosx)/3x2
= limx→a(f(x) – f(a))/(g(x) – g(a))
= limx→a(f(x) – f(a))/(x-a)/(g(x) – g(a))/(x-a)
= (limx→a(f(x) – f(a))/(x-a))/( limx→a (g(x) – g(a))/(x-a))
f'(x) = bx f'(0)
f'(x) = ex