Use formulas (1) and (2) and the power rule to find the derivatives of the following functions. f ( x ) = 1 x 5 . Derivative of Linear Function If f ( x ) = m x + b , then we have f ' ( x ) = m . ( 1 ) Constant Rule The derivative of a constant function f ( x ) = b is zero. That is, f ' ( x ) = 0 . ( 2 ) Power Rule Let r be any number, and let f ( x ) = x r . Then f ' ( x ) = r x r − 1 .
Use formulas (1) and (2) and the power rule to find the derivatives of the following functions. f ( x ) = 1 x 5 . Derivative of Linear Function If f ( x ) = m x + b , then we have f ' ( x ) = m . ( 1 ) Constant Rule The derivative of a constant function f ( x ) = b is zero. That is, f ' ( x ) = 0 . ( 2 ) Power Rule Let r be any number, and let f ( x ) = x r . Then f ' ( x ) = r x r − 1 .
Consider the function f(x) = x²-1.
(a) Find the instantaneous rate of change of f(x) at x=1 using the definition of the derivative.
Show all your steps clearly.
(b) Sketch the graph of f(x) around x = 1. Draw the secant line passing through the points on the
graph where x 1 and x->
1+h (for a small positive value of h, illustrate conceptually). Then,
draw the tangent line to the graph at x=1. Explain how the slope of the tangent line relates to the
value you found in part (a).
(c) In a few sentences, explain what the instantaneous rate of change of f(x) at x = 1 represents in
the context of the graph of f(x). How does the rate of change of this function vary at different
points?
1. The graph of ƒ is given. Use the graph to evaluate each of the following values. If a value does not exist,
state that fact.
и
(a) f'(-5)
(b) f'(-3)
(c) f'(0)
(d) f'(5)
2. Find an equation of the tangent line to the graph of y = g(x) at x = 5 if g(5) = −3 and g'(5)
=
4.
-
3. If an equation of the tangent line to the graph of y = f(x) at the point where x 2 is y = 4x — 5, find ƒ(2)
and f'(2).
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Differential Equation | MIT 18.01SC Single Variable Calculus, Fall 2010; Author: MIT OpenCourseWare;https://www.youtube.com/watch?v=HaOHUfymsuk;License: Standard YouTube License, CC-BY