Graphs of derivatives

Here are a few graphs that should help you understand and better remember the main properties of the first and second order derivative of a function.

There is no exercise related to these but take a little time to play with them.

Limit definition of derivative

Derivative function

Derivative sign

Convex, concave, inflection point

Second order derivative

Second order derivative, convex, concave

Exercises

These are difficult because we haven’t really done the differentiation rules in class. But I think you can figure these out on your own. Just try to follow the rules, use the hints if you get stuck and don’t panic.

You will need the differentiation rules listed in the top table of tab 5.4 Differentiation rules of the lecture notes on derivatives.

For exercises E5 and E6 you need also the table 5.4.2 for the exponential and logarithmic functions.

Exercises E8 and E9 are about the properties of the derivatives as shown in the graphs and as discussed in the class. These are also summarized in tab 5.3 Application of derivatives of the lecture notes on derivatives.

If the hints don’t help and you get completely stuck, slack me.

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E1. Find the derivative function \(f'(x)\) of \(f(x)\) and evaluate the derivative at point \(x = 1\).

\[f(x) = x^2 + 5\]

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E2. Find the derivative function \(f'(x)\) of \(f(x)\) and evaluate the derivative at point \(x = 1\).

\[f(x) = 2x^3 - 6x^2 + 3\]

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E3. Find the derivative function \(f'(x)\) of \(f(x)\). \[f(x) = 2x^{-3} + \frac{x^2}{2}\]

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E4. Find the derivative function \(f'(x)\) of \(f(x)\).

\[f(x) = \sqrt{3x} + 4 \sqrt{x^3} + \frac{1}{\sqrt{x}}\]

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E5. Find the derivative function \(f'(x)\) of \(f(x)\).

\[f(x) = \frac{e^x}{2} + 3\log x\]

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E6. Find the derivative function \(f'(x)\) of \(f(x)\).

\[f(x) = e^{x/2} + \log x^2\]

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E7. Find the derivative function \(f'(x)\) of \(f(x)\).

\[f(x) = (x^2 + 2)(3x^3 - 5x)\]

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E8. Does the function from E1 have a local minima or maxima? Where? Is any of these global?

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E9. Does the function from E2 have a local minima or maxima? Where? Is any of these global?