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Introduction



Math Preliminaries

This section briefly covers the mathematical tools we will use throughout this course. If you can do all the example problems easily, you should be ok skipping this section. If not, watch on.

Part 1. Exponents



Question:
1.1.1. (xa)b=(x^a)^b= ?

Question:
1.1.2. x−1=x^{-1}= ?

Question:
1.1.3. xaxb=x^ax^b=?

Question:
1.1.4. xaxb=\frac{x^a}{x^b}=
Exponents Examples
Simplify the following expressions:
x3x=\frac{x^3}{x}=
(xn)π=(x^n)^\pi=
x1/2x=\frac {x^{1/2}}{x}=
Try these example problems on your own to check your understanding and then watch the video for a walk-through of the answers.


Part 2. Derivatives




Question:
1.1.5. What the symbol for the natural log of x?

Question:
1.1.6. What does f′(x)f'(x) mean?

Question:
1.1.7. What are some different notations for the derivative of y=f(x)y=f(x)?

Question:
1.1.8. What is the derivative of xnx^n

Question:
1.1.9. What is the derivative of ln⁡x\ln x?


Question:
1.1.10. The product rule states the derivative of g(x)b(x)g(x)b(x) is:


Question:
1.1.11. The chain rule states the derivative of g(b(x))g(b(x)) is:
Derivatives Examples
Find the derivatives of the following expressions:
f(x)=3x2f(x)=3x^2
f(x)=x2−10x+4f(x)=x^2-10x+4
f(x)=x1/2f(x)=x^{1/2}
f(x)=ln⁡(1+x2)f(x)=\ln(1+x^2)
Try these example problems on your own to check your understanding and then watch the video for a walk-through of the answers.


Part 3. Partial Derivatives


Question:
1.1.12. What does ∂y∂x\frac {\partial y}{\partial x} mean?

Question:
1.1.13. Given y=f(x,z)y=f(x,z), what are three ways to write the partial derivative of yy with respect to zz?


Question:
1.1.14. How do we treat the other variables in equations when doing partial derivatives?

Question:
1.1.15. What’s the partial derivative of 3xz23xz^2 with respect to x?

Question:
1.1.16. What’s the partial derivative of 3xz23xz^2 with respect to z?
Basic Partial Derivatives Examples
Find all partial derivatives of the following expressions:
f(x,y)=3yx2f(x,y)=3yx^2
f(x,y)=x1/3y2/3f(x,y)=x^{1/3}y^{2/3}
f(x,y)=10ln⁡x+5ln⁡yf(x,y)=10\ln{x} + 5\ln{y}
Try these example problems on your own to check your understanding and then watch the video for a walk-through of the answers.

Advanced Partial Derivatives Example
Find all partial derivatives of the following expressions
f(x,y)=(2x2+3y)(4x+58y2)f(x,y)=(2x^2+3y)(4x+58y^2)
f(x,y)=ln⁡(1+yx2)f(x,y)=\ln(1+yx^2)
Try these example problems on your own to check your understanding and then watch the video for a walk-through of the answers.


Part 4. Solving for unknown variables


Solve for x Example
Solve the following expressions for xx:
x−29ln⁡zy=0x-\frac{29\sqrt{\ln{z}}}{y}=0
x2−y=0x^2-y=0
Try these example problems on your own to check your understanding and then watch the video for a walk-through of the answers.

Note: the following video has an error at minute marker 2:38. It should say y = 1, and therefore x = 1.
Solving a system of equations example
Solve the following expression for xx, yy, and zz
xy=z\frac {x}{y} = z
yx=z\frac {y}{x} = z
2x+4z=102x+4z=10
Use both approaches discussed above to check your understanding and then watch the video for a walk-through of the answers.

For further reading: