1. The Chain Rule
Section 3.6c

2. Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3. 2 8 2
1/3 –3 3 3 –4 5
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
(a)
at x = 2
At x = 2:

3. Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3. 2 8 2
1/3 –3 3 3 –4 5
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
(b)
at x = 3
At x = 3:

4. Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3. 2 8 2
1/3 –3 3 3 –4 5
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
(c)
at x = 3
At x = 3:

5. Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3. 2 8 2
1/3 –3 3 3 –4 5
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
(d)
at x = 2

6. Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3. 2 8 2
1/3 –3 3 3 –4 5
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
(e)
at x = 2

7. Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3. 2 8 2
1/3 –3 3 3 –4 5
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
(f)
at x = 2

8. Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3. 2 8 2
1/3 –3 3 3 –4 5
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
(g)
at x = 3

9. Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3. 2 8 2
1/3 –3 3 3 –4 5
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
(h)
at x = 2

10. Suppose that functions f and g and their derivatives have the
following values at x = 2 and x = 3. 2 8 2
1/3 –3 3 3 –4 5
Evaluate the derivatives with respect to x of the following
combinations at the given value of x.
(h)
at x = 2

11. Slopes of Parametrized Curves
A parametrized curve (x(t), y(t)) is differentiable at t if x and y
are differentiable at t. At a point on a differentiable parametrized
curve where y is also a differentiable function of x, the
derivatives dy/dt, dx/dt, and dy/dx are related by the Chain Rule:
Usually, we write this in a different form…
If all three derivatives exist and ,

12. Practice Problems
Find the equation of the line tangent to the curve at the point
defined by the given value of t.
Find the three derivatives:

13. Practice Problems
Find the equation of the line tangent to the curve at the point
defined by the given value of t.
The line passes through:
And has slope:
Equation of the tangent line:

14. Practice Problems
Find the equation of the line tangent to the curve at the point
defined by the given value of t.
Derivatives:
Point:
Slope:
Equation of the tangent line: