Differentiation Flashcards
Differentiate xn
nxn-1
What is the derivative of ex?
ex
What is the derivative of ekx?
kekx
What is the derivative of ef(x)?
f’(x)ef(x)
What is the derivative of lnx?
1 / x
What is the derivative of ln(ax)?
1 / x
What is the derivative of ln f(x) ?
f’(x) / f(x)
What is the derivative of sinx?
cos x
What is the derivative of sin kx?
kcos kx
What is the derivative of cos x?
-sin x
What is the derivative of cos kx?
-ksin kx
What is the derivative of tanx?
sec2x
What is the derivative of tan kx?
ksec2 kx
What is the derivative of cosec x?
-cosecx cotx
What is the derivative of cosec kx?
-k coseckx cot kx
What is the derivative of sec x?
secx tanx
What is the derivative of sec kx?
kseckx tankx
What is the derivative of cotx?
-cosec2x
What is the derivative of cot kx?
-k cosec2kx
What is the derivative of ax?
ax lna
What is the derivative of akx?
akxk lna
What is the derivative of arcsin x?
1 / √(1-x2)
What is the derivative of arccos x?
-1 / √(1-x2)
What is the derivative of arctan x?
1 / (1+x2)
Find the derivative of sinx from first principles
ONLY in radians
f(x) = sin x
f’(x) = h→0 (f(x+h) -f(x)) / h
f’(x) = h→0 (sin(x+h) - sin(x)) / h
f’(x) = lim h→0 (sinxcosh + cosxsinh - sinx) / h
f’(x) = lim h→0 (sinxcosh - sin x) / h + cosxsinh / h
f’(x) = sinx(cosh - 1) / h + cosx (sinh) / h
(cosh -1) / h → 0 and sinh / h → 1
f’(x) = lim h→0 (sin(x+h)-sinx) / h = cos x
Find the derivative of cosx from first principles?
f(x) = cosx
f’(x) = (cos(x+h) - cosx) / h
f’(x) = h → 0 (cosxccosh - sinxsinh - cosx) / h
f’(x) = h → 0 (cosxcosh - cosx / h) - (sinxsinh / h)
f’(x) = lim h → 0 cosx (cosh-1 / h) - sinx (sinh/h)
cosh -1 /h → 0 and sinh/h → 1
f’(x) = 0-sinx = -sinx
What is the derivative of lnxn?
n lnx
What is the chain rule?
dy/dx = du/dx * dy/du
Do this when there is a function of a function
Use the chain rule to differentiate (x+3)7
u = x + 3 and du/dx = 1
y = u7 and dy/du = 7u6
dy/dx = 7u6 = 7(x+3)6
What is the product rule?
y = uv
dy/dx = (u*dv/dx) + (v*du/dx)
What is the quotient rule?
y = u / v
dy/dx =(v*du/dx - u*dv/dx ) / v2
What is implicit differentiation?
Equations which emphasise x and y as equal partners
d/dx f(y) = f’(y) * dy/dx
e.g. d/dx (y2) = 2y * dy/dx
When is a function concave at a given interval?
When f’‘(x) <= 0 for every value of x in the interval
This is when the gradient is decreasing (maximum)
When is a function convex at an interval?
Only when f’‘(x) >= 0 for every value of x
This is when the gradient is increasing (minimum)
What is a point of inflection in reference to convex and concave?
Point of inflection is where f’‘(x) changes sign
You need to show f’‘(x) = 0 at the point and that there are different signs on either side
Concave on one side and convex on the other
What does the interval [a,b] mean?
a <= x <= b
How do you differentiate parametric equations?
e.g. x = 2at2 and y = 4at
dy/dx = (dy/dt) / (dx/dt)
What does the expression “increasing at the rate of” mean?
Implies differentiation with respect to time
What is a differential equation?
An equation which involves a rate of change
When proportional is mentioned in the question, what should the equations show?
e.g. x is proportional to y
and when decreasing the rate of change (decay)
x = ky
if decay/decrease then x = -ky
