What is the largest the slope of f(x) = sin(x) ever gets? The smallest?
Hint
Remember that "slope" means "derivative."
Answer
This question is asking what is the largest (and smallest) the derivative ever gets. Since f ' (x) = cos(x), the largest the derivative ever gets is 1. The smallest the derivative gets is -1.
Example 2
What is the largest the slope of f(x) = cos(x) ever gets? The smallest?
Hint
Remember that "slope" means "derivative."
Answer
This question is asking what is the largest (and smallest) the derivative ever gets. Since f ' (x) = -sin(x), the largest the derivative ever gets is 1. The smallest the derivative gets is -1.
Example 3
Let f(x) = -sin(x). Based on the discussion in this section, what is a reasonable formula for f ' (x)?
Hint
Look at the graph of f(x) = -sin(x).
Answer
The graph of -sin x looks like this:
We can see its derivative will be 0 at and :
And from the discussion earlier, the largest its slope will be is 1 and the smallest is -1. We can see from the picture where these largest and smallest slopes happen:
This lets us plot some points for the derivative function:
and if we connect the dots in a nice smooth curve, we find the graph of -cos x:
Example 4
Let f(x) = -cos(x). Based on the discussion in this section, what is a reasonable formula for f ' (x)?
Hint
Look at the graph of f(x) = -cos(x).
Answer
The graph of -cos x looks like this:
We can see where its slope is 0, and where its slope is largest and smallest:
We plot some points on the graph of the derivative:
and play connect the dots to find the graph of sin x:
Example 5
Find the derivative of the function. Be sure to use correct notation.
f(x) = sin(x)
Answer
f ' (x) = cos(x)
Example 6
Find the derivative of the function. Be sure to use correct notation.
g(z) = cos(z)
Answer
g ' (z) = -sin(z)
Example 7
Find the derivative of the function. Be sure to use correct notation.
h(y) = -sin(y)
Answer
h ' (y) = -cos(y)
Example 8
Find the derivative of the function. Be sure to use correct notation.
and 
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