2. You can identify where a function increases and decreases by finding the derivative. When you find the derivative, you can plug in a value for x, in order to find the slope of the function at that point.
3. The Chain Rule is a method of find the derivative of a composite function. you begin by working from the outside of the function in. First, you take the derivative of the outside function. Second, you rewrite the interior function. Finally, you multiply by the derivative of the interior function.
f'(x) = sqrt(1 -2x)f'(x) = (1 - 2x)^(1/2)
f'(x) = 1/2(1 - 2x) ^ (-1/2) (-2)
f'(x) = -2/2(1 - 2x)^(1/2)
f'(x) = -1/sqrt(1 - 2x)
Find tangent line at x = -7.5
f(-7.5) = sqrt(1 -2(-7.5)) = sqrt(16)
f(-7.5) = 4
f'(-7.5) = -1 / sqrt ( 1 -2(-7.5))
f'(-7.5) = -1/4
tangent line = y - 4 = -1/4 (x + 7.5)
4. h(x) = f(g(x)) g(-4) = 5 g'(-4) =2 f'(5) = 20. Find h'(-4)
h'(-4) = f'(g(-4))(2)
h'(-4) = f'(5)(2)
h'(-4) =(20)(2)
h'(-4) = 40
