Unizor - Derivatives - Exercises 3

Notes to a video lecture on http://www.unizor.com

Derivatives - Exercise 3

Exercise 3.1
Using the rules of taking a derivative, find the derivative of
f(x) = sin²(x) + cos²(x)
Why the answer is, what it is?

Exercise 3.2
Using the rules of taking a derivative, find the derivative of
f(x) = ln(e^x)
Why the answer is, what it is?

Exercise 3.3
Using the rules of taking a derivative, find the derivative of
f(x) = √x+√x

Answer
[1+1/(2√x)] /2√x+√x

Exercise 3.4
Using the rules of taking a derivative, find the derivative of
f(x) = arctan(1/x)

Answer
−1/(1+x²)

Exercise 3.5
Using the rules of taking a derivative, find the derivative of
f(x) = x^1/x

Answer
x^1/x·(1−ln(x))/x²

Exercise 3.6
Using the rules of taking a derivative, find the derivative of this implicitly defined function
y^x = x^y
(of course, the derivative will also be implicitly defined)

Answer
(y²−x·y·ln(y))/(x²−x·y·ln(x))