1. f'=0 represents the vertex of the original equation because the slope(derivative) is zero, so it is not increasing or decreasing.
2. You can identify where a function increases when you calculate the derivative in front of the vertex and the point is less than the point of the vertex(or critical point). You can tell where it's decreasing if the derivative after the vertex is less than after the vertex(or critical point). If the derivative is negative, the equation is decreasing. If it is positive it is increasing.
3. The chain rule is the process of finding the derivative of a composite function.
If f(x)=g(h(x)) then f'(x)=g'(h(x))*h'(x)
In reality all your doing is...
1. Taking the derivative of the outside function
2. Rewriting the interior function
3. Multiplying by the derivative of the inside
Find the derivative and the equation of the tangent line where x=2.
f(x)= (2x^2 - x)^2 f'= 2(2x^2 - x) * 4x-1 = 2(4x-1)(2x^2 - x)
(8x^3 - 6x^2 + x)2 f'=16x^3 - 12x^2 + 2x
Point: (2, 36) Slope:84 Equation: y= 84x + 204
4. h(x)= f(g(x)) g(-4)=5 g'(-4)=2 f'(5)=20 Find h'(-4).
