# Gradients and Directional Derivatives - PowerPoint by 62I0JCj

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Derivatives

Chp 15.6
Putting it all together…
• Over the past several classes you have
learned how to use and take partial
derivatives
• Today we look at the essential difference
between defining slope along a 2D curve
and what it means on a 3D surface or
curve
 x2
10e y
Example…What’s the slope of f ( x, y )                   at (0,1/2)?
1  x2  y 2

What’s wrong with the way
the question is posed?

What’s the slope along                          What’s the slope
the direction of the x-                         along the direction of
axis?                                           the y-axis?
Quick estimate from the contour
plot:

z  (0  4)  4
y  (0.5  0)  0.5            z
m  8

Look at this in Excel:
The Directional Derivative
• We need to specify the direction in which the
change occurs…
• Define, via a slightly modified Newton quotient:

Du f ( x, y)  f x ( x, y)a  f y ( x, y)b
• This specifies the change in the direction of the
vector u = <a,b>
• We can write the Directional derivative as:

Du f ( x, y, z )  f x , f y , f z   u1 , u2 , u3