9: Differentiation Flashcards

1
Q

What is the result when you differentiate sin(x)?

A

cos(x)

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2
Q

What is the result when you differentiate cos(x)?

A

-sin(x)

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3
Q

Differentiate sin(x) from first principles.

A
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4
Q

Differentiate cos(x) from first principles.

A
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5
Q

y = sin(kx)

dy/dx =

A

kcos(kx)

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6
Q

y = cos(kx)

dy/dx =

A

-ksin(kx)

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7
Q

y = ekx

dy/dx =

A

kekx

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8
Q

y = ln(x)

dy/dx =

A

1/x

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9
Q

y = ln(kx)

dy/dx =

A

y = ln(kx) = ln(k) + ln(x)

ln(k) is a constant

dy/dx = 1/x

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10
Q

y = akx

dy/dx =

A

akxk ln(a)

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11
Q

y = akx and dy/dx = akx kln(a)

When is this true?

A

K ϵ ℝ, a > 0

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12
Q

Show that the derivative of ax is axln(a)

A

y = ax

ln(y) = ln(ax) = xln(a)

y = exln(a)

Where y = ekx, dy/dx = kekx

dy/dx = ln(a) exln(a) = ln(a) eln(a^x)

= ln(a) ax = ax ln(a)

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