BC Calculus: §3.6 Chain Rule
Given the following tabular information about differentiable functions \(f(x)\) and \(g(x) \text{ at }x = 2\) and \(x = 3\):
x | \(f(x)\) | \(g(x)\) | \(f'(x)\) | \(g'(x)\)
|
2 | 8 | 2 | \(\frac{1}{3}\) | -3
|
3 | 3 | -4 | e | 5
|
Determine the value of \(\frac{d}{dx}\) of:
- \(2f(x) \text{ at }x = 2\)
- \(f(x) + g(x) \text{ at }x = 3\)
- \(f(x) \cdot g(x) \text{ at }x = \)3
- \( \frac{f(x)}{g(x)} \text{ at }x = 2\)
- \(f(g(x)) \text{ at }x = 2\)
Given \(f(x)\) and \(g(x)\) are differentiable functions over the real numbers, \(f'(x) = - f(x)\) and \(g'(x) = g(x) + 1\), and:
x | \(f(x)\) | \(g(x)\)
|
-3 | 2 | 1
|
-2 | 1 | 2
|
-1 | 3 | 0
|
0 | -1 | 3
|
1 | -2 | -1
|
2 | 0 | -2
|
Evaluate \(\frac{d}{dx}\) of the following:
- \(f(g(x)) \text{ at }x = -1\)
- \(g(f(x)) \text{ at }x = -2\)
- \(f(x)\cdot g(x - 1) \text{ at }x = 1\)
- \(\frac{g(x)}{f(x+5)} \text{ at }x = -3\)