Differentiation Flashcards

1
Q

Differentiate xn

A

nxn-1

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2
Q

What is the derivative of ex?

A

ex

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3
Q

What is the derivative of ekx?

A

kekx

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4
Q

What is the derivative of ef(x)?

A

f’(x)ef(x)

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5
Q

What is the derivative of lnx?

A

1 / x

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6
Q

What is the derivative of ln(ax)?

A

1 / x

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7
Q

What is the derivative of ln f(x) ?

A

f’(x) / f(x)

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8
Q

What is the derivative of sinx?

A

cos x

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9
Q

What is the derivative of sin kx?

A

kcos kx

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10
Q

What is the derivative of cos x?

A

-sin x

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11
Q

What is the derivative of cos kx?

A

-ksin kx

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12
Q

What is the derivative of tanx?

A

sec2x

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13
Q

What is the derivative of tan kx?

A

ksec2 kx

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14
Q

What is the derivative of cosec x?

A

-cosecx cotx

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15
Q

What is the derivative of cosec kx?

A

-k coseckx cot kx

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16
Q

What is the derivative of sec x?

A

secx tanx

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17
Q

What is the derivative of sec kx?

A

kseckx tankx

18
Q

What is the derivative of cotx?

19
Q

What is the derivative of cot kx?

A

-k cosec2kx

20
Q

What is the derivative of ax?

21
Q

What is the derivative of akx?

22
Q

What is the derivative of arcsin x?

A

1 / √(1-x2)

23
Q

What is the derivative of arccos x?

A

-1 / √(1-x2)

24
Q

What is the derivative of arctan x?

A

1 / (1+x2)

25
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
26
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
27
What is the derivative of lnxn?
n lnx
28
What is the chain rule?
dy/dx = du/dx \* dy/du Do this when there is a function of a function
29
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
30
What is the product rule?
y = uv dy/dx = (u\*dv/dx) + (v\*du/dx)
31
What is the quotient rule?
y = u / v dy/dx =(v\*du/dx - u\*dv/dx ) / v2
32
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
33
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)
34
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)
35
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
36
What does the interval [a,b] mean?
a \<= x \<= b
37
How do you differentiate parametric equations? e.g. x = 2at2 and y = 4at
dy/dx = (dy/dt) / (dx/dt)
38
What does the expression "increasing at the rate of" mean?
Implies differentiation with respect to time
39
What is a differential equation?
An equation which involves a rate of change
40
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