First Derivative Formulas
y is a function y = y(x)
C = constant, the derivative(y') of a constant is 0
y = C => y' = 0
example: y = 5, y' = 0
If y is a function of a type y = x^{n}
the formula for the derivative is:
y = x^{n} => y' = nx^{n1}
example: y = x^{3} y' = 3x^{31} = 3x^{2}
y = x^{3} y' = 3x^{4}
From the upper formula we can say for derivative y' of a function y = x = x^{1} that:
if y = x then y'=1
y = f_{1}(x) + f_{2}(x) + f_{3}(x) ...=>
y' = f'_{1}(x) + f'_{2}(x) + f'_{3}(x) ...
This formula represents the derivative of a function that is sum of functions.
example: If we have two functions f(x) = x^{2} + x + 1 and
g(x) = x^{5} + 7 and y = f(x) + g(x) then y' = f'(x) + g'(x) =>
y' = (x^{2} + x + 1)' + (x^{5} + 7)' = 2x^{1} + 1 + 0 + 5x^{4} + 0 = 5x^{4} + 2x + 1
If a function is multiplication of two functions the derivate formula is:
y = f(x).g(x) => y' = f'(x)g(x) + f(x)g'(x)
If f(x) = C(C is a contstant), and y = f(x)g(x)
y = Cg(x) y'=C'.g(x) + C.g'(x) = 0 + C.g'(x) = C.g'(x)
y = Cf(x) => y' = C.f'(x)
There are examples of the following formulas in the task section.
y = ln x => y' = ^{1}/_{x}
y = e^{x} => y' = e^{x}
y = sin x => y' = cos x
y = cos x => y' = sin x
y = tan x => y' = ^{1}/_{cos2x}
y = cot x => y' = ^{1}/_{sin2x}
if a function is a function of function: u = u(x)
y = f(u) => y' = f'(u).u'
example: let's have function y = sin(x^{2})
in this situation u = x^{2}, f(u) = sin(u), the derivatives are f'(u) = cos(u), u' = 2x
y' = (sin(u))'.u' = cos(x^{2}).2x = 2.x.cos(x^{2})
Derivative Problems
1) f(x) = 10x + 4y, what will be the first derivative f'(x) = ?
ANSWER: We can use the formula for the derivate of function that is sum of
functions
f(x) = f_{1}(x) + f_{2}(x), f_{1}(x) = 10x, f_{2}(x) = 4y
for the function f_{2}(x) = 4y, y is a constant because the argument of f_{2}(x) is x
so f'_{2}(x) = (4y)' = 0. So the derivative function of f(x) is: f'(x) = 10 + 0 = 10.
2) Calculate the derivative of f(x) = 

ANSWER:
we have two functions h(x) = x^{10} and g(x) = 4.15 + cos x
the function f(x) is h(x) divided by g(x). h'(x) = 10x^{9} g'(x) = 0  sin x = sin x
f'(x) = 
h'(x).g(x)  h(x).g'(x) 
(g(x))^{2} 

f'(x) = 
10x^{9}(4.15 + cos x)  x^{10}(sin x) 
(4.15 + cosx)^{2} 

= 
x^{10}sin x + 10(60 + cos x)x^{9} 
(60 + cosx)^{2} 

3) f(x) = ln(sinx). what is the derivative of the function f(x)?
ANSWER: To solve the task we have to use the last formula.
As we can see f(x) is a function of function of function
f(x) = h(g(x)) where h = ln, and g = sin x
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