9. Differentiation Flashcards

1
Q

Concave functions

A

The second derivative is <= 0 for all x, any two points on the curve joined by a line are below the curve

Rise to a maximum and then fall again

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2
Q

Convex functions

A

The second derivative is >= 0 for all x, any two points on the curve joined by a line are above the curve

Fall to a minimum and then rise again

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3
Q

Chain rule process

A
  1. State the original function
  2. State what your temporary variable is (u = inside of brackets)
  3. Set y to u^power and do dy/du
  4. Find du/dx
  5. dy/dx = dy/du x du/dx and replace u with the inside of the brackets
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4
Q

Chain rule f(x)

A

Write y = f(x) and then put f’(x) at the end

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5
Q

sin(x) differentiated

A

cos(x)

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6
Q

cos(x) differentiated

A

-sin(x)

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7
Q

Differentiating sin(x) and cos(x) by first principles

A

Use the angle addition formulae and use sin(h)/h and 1-cos(x)/h = 0 by finding them with small angles

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8
Q

e^x differentiated

A

e^x

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9
Q

ln(x) differentiated

A

1/x

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10
Q

sin(h)/h

A

1

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11
Q

1 - cos(h)/h

A

0

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12
Q

Product rule

A

Used where y = uv where u and v are two functions

  1. Differentiate each with respect to x
  2. Multiply by the other function
  3. Sum
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13
Q

e^(ax + b) differentiated

A

a x e^(ax+b)

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14
Q

ln(x^2) differentiated

A

2x

x^2

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15
Q

ln(ax+b) differentiated

A

a
———
ax + b

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16
Q

cos(ax) differentiated

A

-a sin (ax)

17
Q

sin(ax) differentiated

18
Q

Quotient Rule uses

A

where y can be written as u/v where u,v are functions

19
Q

Quotient rule

A
.      du         dv
    v ----  -  u -----
dy   dx         dx
---- = ----------------
dx            v^2
20
Q

Tan x differentiated

21
Q

Explicit Equations

A

Where y can be written as f(x)

22
Q

Implicit equations

A

Not written in the form y = f(x)

E.g. x^2 + y^2 = 5, y + 7 = 3x

23
Q

d/dx (f(y))

A

f’(y) x dy/dx

dy/dx won’t be in terms of numbers

24
Q

Implicit differentiation steps

A
  1. Write d/dx(LHS) = d/dx(RHS)
  2. Differentiate each side leaving the derivative of y as dy/dx
  3. Rearrange and factor out dy/dx
  4. Divide for dy/dx=
25
d/dx(xy)
y + x(dy/dx)
26
d/dx a^(bx+d)
b ln(a) a^bx+d
27
sec(x) differentiated
sec(x)tan(x)
28
cosec(x) differentiated
-cosec(x) cot(x)
29
y = f(g(x)) differentiated (chain rule)
g’(x) f’(g(x))
30
Differentiating inverse trig
Rearrange for x = a function of y Find dx/dy Find the reciprocal for dy/dx Substitute to be in terms of x
31
Connected rates of change
Use the chain rule to write the derivative you need as a product of two derivatives You may need to differentiate an equation for area or volume and do 1/ a derivative
32
a^x differentiated
ln(a) a^x