Gradients and Directional
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
Example…What’s the slope of
f ( x, y ) 
10 e
x
2
y
1 x  y
2
2
at (0,1/2)?
What’s wrong with the way
the question is posed?
What’s the slope along
the direction of the xaxis?
What’s the slope
along the direction of
the y-axis?
Quick estimate from the contour
plot:
 z  (0  4 )   4
 y  (0 .5  0 )  0 .5
m  8
Look at this in Excel:
z
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>
The Gradient
• We can write the Directional derivative as:
Du f ( x , y , z ) 
Gradient of f(x,y,z)
fx, fy, fz
u1 , u 2 , u 3
 f ( x, y, z )
A Key Theorem
• Pg 982 – the Directional derivative is
maximum when it is in the same direction
as the gradient vector!
• Example: If your ski begins to slide down
a ski slope, it will trace out the gradient for
that surface!