Concave functions
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
Convex functions
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
Chain rule process
- State the original function
- State what your temporary variable is (u = inside of brackets)
- Set y to u^power and do dy/du
- Find du/dx
- dy/dx = dy/du x du/dx and replace u with the inside of the brackets
Chain rule f(x)
Write y = f(x) and then put f’(x) at the end
sin(x) differentiated
cos(x)
cos(x) differentiated
-sin(x)
Differentiating sin(x) and cos(x) by first principles
Use the angle addition formulae and use sin(h)/h and 1-cos(x)/h = 0 by finding them with small angles
e^x differentiated
e^x
ln(x) differentiated
1/x
sin(h)/h
1
1 - cos(h)/h
0
Product rule
Used where y = uv where u and v are two functions
- Differentiate each with respect to x
- Multiply by the other function
- Sum
e^(ax + b) differentiated
a x e^(ax+b)
ln(x^2) differentiated
2x
—
x^2
ln(ax+b) differentiated
a
———
ax + b
cos(ax) differentiated
-a sin (ax)
sin(ax) differentiated
a cos(ax)
Quotient Rule uses
where y can be written as u/v where u,v are functions
Quotient rule
. du dv
v ---- - u -----
dy dx dx
---- = ----------------
dx v^2Tan x differentiated
Sec^2x
Explicit Equations
Where y can be written as f(x)
Implicit equations
Not written in the form y = f(x)
E.g. x^2 + y^2 = 5, y + 7 = 3x
d/dx (f(y))
f’(y) x dy/dx
dy/dx won’t be in terms of numbers
Implicit differentiation steps
- Write d/dx(LHS) = d/dx(RHS)
- Differentiate each side leaving the derivative of y as dy/dx
- Rearrange and factor out dy/dx
- Divide for dy/dx=
