1. Calculus Review

2. Slope Slope = rise/run = Dy/Dx = (y2 – y1)/(x2 – x1)
Order of points 1 and 2 not critical
Points may lie in any quadrant: slope will work out
Leibniz notation for derivative based on Dy/Dx; the derivative is written dy/dx

3. Exponents
x0 = 1

4. Derivative of a line y = mx + b slope m and y axis intercept b
derivative of y = axn + b with respect to x:
dy/dx = a n x(n-1)
Because b is a constant -- think of it as bx0 -- its derivative is 0b-1 = 0
For a straight line, a = m and n = 1 so
dy/dx = m 1 x(0), or because x0 = 1,
dy/dx = m

5. Derivative of a polynomial
In differential Calculus, we consider the slopes of curves rather than straight lines
For polynomial y = axn + bxp + cxq + …
derivative with respect to x is
dy/dx = a n x(n-1) + b p x(p-1) + c q x(q-1) + …

6. Example y = axn + bxp + cxq + … dy/dx = a n x(n-1) + b p x(p-1) +
c q x(q-1) + …

7. Numerical Derivatives
slope between points

8. Derivative of Sine and Cosine
period of both sine and cosine is 2p
d(sin(x))/dx = cos(x)
d(cos(x))/dx = -sin(x)

9. Partial Derivatives Functions of more than one variable
Example: h(x,y) = x4 + y3 + xy

10. Partial Derivatives
Partial derivative of h with respect to x at a y location y0
Notation dh/dx|y=y0
Treat ys as constants
If these constants stand alone, they drop out of the result
If they are in multiplicative terms involving x, they are retained as constants

11. Partial Derivatives Example: h(x,y) = x4 + y3 + xy
dh/dx|y=y0 = 4x3 + y0

12. WHY?

13. Gradients
del C (or grad C)
Diffusion (Fick’s 1st Law):

14. Numerical Derivatives
slope between points
MATLAB
c=[];
[dcdx,dcdy]=gradient(c)
contour([1:20],[1:20],c)
hold
quiver([1:20],[1:20],-dcdx,-dcdy)

15. Mathematica

16. Mathematica