Y2, C9 - Differentiation Flashcards
Differential of cos(x)
-sin(x)
Differential of sin(x)
cos(x)
Derivative of sin(kx)
kcos(kx)
Derivative of cos(kx)
-ksin(kx)
Differential of a^x
lna * a^x
Differential of a^kx
klna * a^kx
Differential of lnx
1 / x
Differential of ln(kx)
1 / x
How do you differentiate a composite function (f(g(x))
Chain rule
How do you differentiate the product of two functions (y = x * sin2x)
Product rule
How do you differentiate the division (or quotient) of two functions
The quotient rule
What is the chain rule
dy / dx = (dy / du) * (du / dx)
What is the shortcut of the chain rule
Differentiate as usual
Multiply by the differential of u
Use the shortcut of the chain rule to differentiate (3x^4 + x)^5
5(3x^4 + x)^4 * (12x^3 +1)
Mentally differentiate 3(8 - x)^-6
-18(8 - x)^-7 * (-1) =
18(8 - x)^-7
Mentally differentiate e^(x^2 + x)
e^(x^2 + x) * (2x + 1)
Mentally differentiate (2^x + 1)^2
2(2^x + 1) * (ln2 * 2^x)
Mentally differentiate y = -sin^-2(x)
y = - (sinx)^-2
–> 2sin^-3(x) * cosx
–> 2cos(x)sin^-3(x)
–> 2cos(x)cosec^3(x)
Differentiate ln(x^3)
3x^2 / x^3
3 / x
What is the reciprocal of dy / dx
1 / (dx / dy)
What is dy / dx when x = 2y^2 + y
dx / dy = 1 / (dy / dx)
dx / dy = 4y + 1
dx / dy = 1 / (4y + 1)
What is the product rule
If y = uv then dy/dx = u(dv/dx) + v(du/dx)
OR
y’ = uv’ + vu’
Differentiate ln(3x)
1 / x
Differential of ae^bx
abe^bx
What is the quotient rule
If y = u / v, then dy / dx =
(vu’ - uv’) / v^2
Find the stationary point of y = sinx / e^2x
Quotient rule: dy / dx = (cosx - 2sinx) / e^2x
Multiply out by e^2x (e^2x = 0 has no solutions)
Divide remaining terms by tanx
1 - 2tanx = 0
1/2 = tanx
x = 0.464, y = 0.177
What is the differential of tan(kx)
k sec^2(kx)
What is the differential of sec(kx)
k sec(kx) * tan(kx)
What rule would you use to differentiate tan, sec, cot, cosec
Quotient rule
What is the differential of cot(x)
-cosec^2(x)
What is the differential of cosec(x)
-cosec(x) * cot(x)
What is 1 + tan^2(x) equal to
sec^2(x)
Given that x = 2 sin(y), express dy/dx in terms of x
dx / dy = 2cos(y)
dy / dx = 1 / 2cos(y)
cos(y) = (root(4-x^2) / 2)
Sub in for cos(y)
dy / dx = 1 / (root(4-x^2))
What is the equation for parametric differentiation
dy /dx = (dy / dt) / (dx / dt)
What is the difference between explicit and implicit functions
Explicit functions have a variable as the subject
Implicit functions do NOT have a variable as the subject
Differentiate y^2 with respect to x
= 2y * dy/dx
In general when differentiating a function of y, but with respect to x, what should you multiply by
dy / dx
Differentiate x^2 + cos(y) with respect to x
2x - siny * (dy/dx)
Find dy/dx in terms of x and y where x^3 + x + y^3 + 3y = 6
3x^2 + 1 + (3y^2 + 3)dy/dx = 0
dy/dx = (-3x^2 - 1) / (3y^2 + 3)
What does it mean if a tangent is parallel to the x axis
The gradient is 0
dy/dx = num / denom
num = 0
What does it mean if the tangent is parallel to the y axis
The gradient is undefined
dy/dx = num / denom
denom = 0
What way is a curve swerving if it is concave
Right
What way is a curve swerving if it is convex
Left
What is f’‘(x) equal to at a point of inflection
0
When is f(x) convex
f’‘(x) > 0
When is f(x) concave
f’‘(x) < 0
What does the rate of something mean
How it changes per time (seconds)
What would the unit of measurement be for dA / dt where t is time and A is area
cm^2 * s^-1
How would you find the equation for the rate of change of volume of a sphere if you know the rate of change of radius
dV / dt = (dV/dr) * (dr/dt)
V = 4/3 * pi * r^3
dV/dr = 4 * pi * r^2
Therefore dV/dt = 4pi*r^2 * dr/dt
If you know V in terms of r, S in terms of r and dV/dt. How would you calculate dS/dt.
V = volume
S = surface area
r = volume
Find dV/dr, dS/dr, dV/dt
dS/dt = (dS/dr) * (dr/dV) * (dV/dt)
Sub in known values and find dS/dt in terms of r
How do you solve differential equations
Getting y in terms of x with NO dy/dx
Get y on LHS (possibly factorising first)
Then integrate both sides
Find the general solution to dy/dx = xy + y
dy/dx = y(x+1)
(1/y) * (dy/dx) = x + 1
(1/y) dy = (x+1) dx
lny = 0.5x^2 + x + c
y = Ae^0.5x^2 + x
where A = e^c
When solving differential equations, if your ‘+c’ becomes an ‘ln’, what should you call it
ln k
How do you solve differential equations with boundary conditions
Sub in the given values to find the value of ‘c’ and then rewrite in general solution
The rate of increase of a population is proportional to current population, form a differential equation and find its general solution
dP/dt = kP
(1/P) * dP/dt = k
ln(P) = kt + c
P = Ae^kt
