Implicit Differentiation

Implicit Differentiation and derivatives with logs

By Jeff Parent

Reasons a f'(a) COULD FAIL TO EXIST

Three reasons why f'(x) could fail to exist at the point x=a

1. The function is discontinuous at a point "a"

2. The function has a vertical tangent at "a".

3. The function has a "corner point" at "a"

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The function is discontinuous at a point.​

The function could have a "removable discontinuity", a "vertical asymptote", a "jump discontinuity" or simply be "undefined" at a x="a". If f(a) DNE (does not exist), then f'(a) does not exist either. If f(a) is discontinuous, then f'(a) fails to exist.

Subject | Subject

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Multiple Choice

How many places are there where dy/dx = 0 OR is undefined on the circle shown as part of example F?

• 1

• 2

• 3

• 4

Multiple Choice

The one thing that the instructor said about logs and derivatives that was a "strong suggestion" was to

• Always use logarithmic differentiation

• Use the rules of logarithms to simplify things before you differentiate

• Use the rules of logarithms to simplify things after you differentiate

• Multiply both sides of the equation by the lowest common denominator to clear out any fractions

Multiple Choice

At what time(s) is the particle at rest when it's position function given by $$s\left(t\right)=t^3+9t^2+15t$$  

• t=0

• t=1

• t=5

• t=1 and t=5

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f(a) has a​ vertical tangent at "a". For the graph at the left, the vertical tangent is at x=0​

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The function has a "corner" point, like the one shown at the right at x=3​

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