Derivatives
What is the definition of the derivative?
f'(x) = lim (h→0) [f(x+h) - f(x)] / h.
the set of all possible input values (x-values) for which the function is defined
What is the domain of a function?
What is the derivative of 1/(4√x)
(-1)/(4 (4√x5))
equations that express the coordinates x and y as separate functions of a third variable, usually t
What are parametric equations?
Find the derivative of h(x)=sin25x
Needed to use chain rule
h'(x) = 5sin(10x)
a function formed by applying one function to the result of another, denoted (f∘g)(x)=f(g(x))
What is a composite function?
using the definition of the derivative, find the derivative of 3x2 + 2x4 ? Must show work to get credit!
Compute f(x+h) x+h into the function.
Subtract f(x) from f(x+h) — this gives the numerator f(x+h)−f(x)
Simplify the expression f(x+h)−f(x) completely.
Divide the result by h.
Take the limit as h→0.
Write the final derivative result: 8x3+6x
there are no breaks, jumps, or holes in the graph at that point
What does it mean for a function to be continuous at a point?
Using the definition of the derivative, find the derivative of (x)/(x +1)
Compute f(x+h) by substituting x+hx+h into the function.
Form the difference f(x+h)−f(x)
Combine the two rational expressions using a common denominator.
Simplify the result as much as possible.
Divide the entire expression by h.
Take the limit as h→0
(1)/((x+1)^2)
What is a function expressed as the ratio of two polynomials, like (P(x))/(Q(x)) where Q(x)≠0
What is a rational function?