MST124 BOOK C UNIT 7

Derivative of Sine
- when x = 0, the graident of the sine graph seems to be about the same as the gradient of the line y = x (about 1)
- as x increases, the graph gradually gets less steep
- the gradient gradually decreases unit x = pi/2, when the gradient seems to be 0
- the gradient decreases slowly when x is only a little larger than 0, but decreases more rapidly as x gets closer to pi/2 
- for x between 0 and pi/2 the value of the derivative seems to decrease from 1 to 0, slowly at first, but then more rapidly
- between pi/2 and pi, the gradient starts at 0, and then it becomes negative
- at first the graph is not very steep (small -ve values) but it becomes steeper (larger mag of -ve values)
- when x = pi, the gradient seems to be about the same as the line y = -x (about -1)
- the gradient fo the line decreases fairly rapidly when x is only slightly larger than pi/2, but decreases more slowly as x gets closer to pi 
- for x between pi/2 and pi, the value of the derivative seems to decrease from 0 to -1, slowly at first and then more rapidly 

Sine (cont.)
- the graph of the sine function repeats every 2*pi units, so its gradients also repeat every 2*pi units
- the graph of the derivative of the sine function looks like the cosine graph
- if f(x) = sin(x), then f'(x) = cos(x) 

Derivative of Cosine 
- the graph of the derivative cos(x) looks like the sine function graph, reflected in the x-axis
- y = -sin(x)
- if f(x) = cos(x), then f'(x) = -sin(x) 

Derivative of Tangent
- if f(x) = tan(x), then f'(x) = sec²(x) 
or 1/cos²(x) 
- the gradients of y = tan(x) are always +ve and the graph of y = tan(x) is the least steep whenever x is a multiple of pi 

Derivative of exp
- the derivative of the exponential function is the exponential function
f(x) = e^x and f'(x) = e^x 

Derivative of ln
- this function is the inverse of the exponential function f(x) = e^x
- its graph is the reflection of the exponential function in the line y = x 
- if f(x) = lnx then f'(x) = 1/x 

Product rule (Lagrange notation)
If k(x) = f(x)g(x), then
k'(x) = f(x)g'(x) + f'(x)g(x)
for all values of x at which both f and g are differentiable 

Product rule (Leibniz notation)
If y = uv, where u and v are functions of x, then 
dy/dx = u(dv/dx) + v(du/dx)
for all values of x at which both u and v are differentiable 

Product rule (informal)
(derivative of product) = (first) x (derivative of second) + (second) x (derivative of first) 

Quotient rule (Lagrange notation)
If k(x) = f(x)/g(x), then
k'(x) = (g(x)f'(x) - f(x)g'(x))/(g(x))²

Quotient rule (Leibniz notation)
If y = u/v, where u and v are functions of x, then
dy/dx = (v(du/dx) - u(dv/dx))/v²

Quotient rule (informal)
(derivative of quotient) = ((bottom) x (derivative of top) - (top) x (derivative of bottom)) / (bottom)² 

Derivatives of cosec, sec and cot
d/dx(cosec(x)) = -cosec(x)cot(x)
d/dx(sec(x)) = sec(x)tan(x)
d/dx(cot(x)) = -cosec²(x) 

Chain rule (Leibniz notation) 
If y is a function of u, where u is a function of x, then
dy/dx = dy/du*du/dx
for all values of x where y as a function of u, and u as a function of x are differentiable 

(EXAMPLE 7 PAGE 26 BOOK C)

Chain rule (Lagrange notation)
If k(x) = g(f(x)), where f and g are functions, then k'(x) = g'(f(x))f'(x) 
for all values of x such that f is differentiable at x and g is differentiable at f(x) 

Derivative of a function of the form k(x) = f(ax)
If k(x) = f(ax), where a is a non-zero constant, then
k'(x) =af'(ax)
for all values of x such that f is differentiable at ax

Derivative of a function of the form k(x) = f(ax + b)
If k(x) = f(ax + b), where a and b are constants with a non-zero, then
k'(x) = af'(ax + b) 
for all values of x such that f is differentiable at ax + b

Checklist for differentiating a function
1 - is it a standard function (derivative in Handbook)?
2 - Can you use the constant multiple rule and/or sum rule?
3 - Can you rewrite it to make it easier to differentiate? (e.g., multiply out brackets)
4 - Is it of the for f(ax) or f(ax + b) where a and b are constants? (use rules on this page)
5 - Can you use the product rule? (f(x) = something x something)
6 - Can you use the quotient rule? (f(x) = something/something)
7 - Can you use the chain rule? (f(x) = a funciton of something 

Strategy
To solve an optimisation problem
1 - identify the quantity that you can change, and represent it by a variable, noting the possible values that it can take
- identify the quantity to be maximised or minimised, and represent it by a variable
- these variables are the independent and dependent variables, respectively 
2 - find a formula for the dependent variable in terms of the independent variable 
3 - use the techniques of differential calculus to find the value of the independent variable that gives the maximum/minimum value of the dependent variable
4 - interpret your answer in the context of the problem 

Inverse function rule (Leibniz notation)
If y is an invertible function of x, then
dy/dx = 1 / dx/dy 
for all values of x such that dx/dy exists and is non zero 

d/dx(sin^-1x) = 1/(sqrt(1-x²))
d/dx(cos^-1x) = - 1/(sqrt 1-x²))
d/dx(tan^-1x) = 1/(1+x²) 

Inverse function rule (Lagrange notation)
If f is a function with inverse function f^-1, then (f^-1)'(x) = 1/(f'(f^-1(x)))
for all values of x such that f'(f^-1(x)) exists and is non-zero 

- table of standard derivatives can be found in the handbook 

- Any function of the form F(x) = x² + c, where c is a constant, has derivative F'(x) = 2x
- so each function of the form F(x) = x² + c is an antiderivative of the function f(x) = 2x 
- the formula F(x) = x² + c describes the complete family of antiderivatives of the function f(x) = 2x
- it is known as the indefinite integral of the function f(x) = 2x

- if f is a function that has an antiderivative, and f is continuous, then the general function
F(x) = (formula for any particular antiderivative of f) + c
where c represents any constant, and whose domain is the same as the domain of f, describes the complete family of antiderivatives of f 
- F is the indefinite integral of the function f 

Antiderivatives and Indefinite integrals
Suppose that f is a function
An antiderivative of f is any specific function whose derivative is f
If f has an antiderivative, and f is continuous, then the indefinite integral of f is the general function obtained by adding an arbitrary constant c to the formula for an antiderivative of f. It describes the complete family of antiderivatives of f. 

!!(You differentiate F(x) to get f(x))!!

Indefinite integral of a power function
For any number n except n = -1
the indefinite integral of xⁿ is (1/n+1)*xⁿ⁺¹ + c

Constant multiple rule for antiderivatives
if F(x) is an antiderivative of f(x), and k is a constant, then kF(x) is an antiderivative of kf(x)

Sum rule for antiderivatives
If F(x) and G(x) are antiderivatives of  f(x) and g(x), then F(x) + G(x) is an antiderivative of f(x) + g(x) 

Indefinite integral of a constant function
The indefinite integral of the constant a is ax + c

- the amount of the change is the same for all the antiderivatives of f
- the graphs of all the antiderivatives are vertical translations of each other

- suppose that f is a function that has an antiderivative, and f is continuous
- if a and b are numbers in the domain of f, then all the antiderivatives of f change by the same amount as x changes from x = a to x = b
- the value of F(b) - F(a) is the same for every antiderivative F of f 

Indefinite integral of the reciprocal function
The indefinite integral of 1/x is ln(modulus of x) + c
If x takes only +ve values, then the indefinite integral of 1/x is simply lnx + c

- antiderivative of cos(x) is sin(x)
- antiderivative of -sin(x) is cos(x)
- antiderivative of sin(x) is -cos(x)
- antiderivative of e^x is e^x