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.
Use the sum rule to decompose the function into 2 terms and then get the derivative of each term separately.
You evaluate the derivative at a point as any other function.
Once you split the function into two terms around the + sign, use the rule for differentiating a constant (for \(f(x) = c\), where \(c\) is any number, \(f'(x) = 0\)) and a power (for \(f(x) = x^n\), where \(n\) is any number, \(f'(x) = n x^{n-1}\)). Then put the two terms back together with the + sign.
General strategy: split the fucntion into terms around the + and - sings (sum, difference rule), differentiatiate each term seperately, then put all the terms back together with + and - sings.
The first term after the split around the + and - sings is \(2x^3\). Here \(c = 2\) (a constant), \(g(x) = x^3\). You use the constant multiple rule for \(c g(x)\) and the power rule for \(g(x)\).
Don’t panic. There is nothing wrong with \(n = -3\) in the power rule and \(c = \frac{1}{2}\) in the constant multiple rule. Just use them as usually.
Remember that you can write the square root \(\sqrt{x}\) as a power? And the same goes for \(\frac{1}{x}\).
Check out the exponent and logarithm differentiation rules. Then apply the constant multiple rule.
Look at the rules in table 5.4.2 with exponential and logarithm operating over \(g(x)\). Then think well what is \(g(x)\) and apply tue rules.
Use the product rule.
In the product rule, you need to decide how to split the fucntion \(f(x)\) into a product of 2 terms \(h(x)\) and \(g(x)\) as \(f(x) = g(x) h(x)\).
Once you decide on the split, you get the derivatives \(h'(x)\) and \(g'(x)\) seperately.
Finally you put them back together following the product rule: \(f'(x) = g'(x) h(x) + g(x) h'(x)\).
(It’s a good idea to simplify the final result.)
Think about the properties of the first and second derivatives? Any of these can help us to find the minima and maxima?
Start from the first derivative and find the critical points.
Then decide whethere these are max or mins either looking at the values of the function at these points and some points around, or by using the second derivative rule.