Introduction to Derivatives

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Introduction to Derivatives Problems for "The Concept of the Derivative"

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Topics Introduction to Derivatives Problems for "The Concept of the Derivative"

Problem : f (x) = 2x. Find f'(x)

f'(x)		=
		=
		=
		= 2
		= 2

Problem : f (x) = - x². What is the f'(x)?

f'(x)		=
		=
		=
		= - (2x + h)
		= - 2x

Problem : Find the slope of the line perpendicular to the tangent of the graph of f (x) = - x² at x = 1.

f'(1) = - 2.
This is the slope of the tangent. The perpendicular to the tangent has a slope that is the negative reciprocal of this, which is

.

Problem : Find f'(1) for f (x) = - x² without solving for f'(x) first.

If we are trying to find the derivative at a single point only, we can substitute that point directly into the limit of the difference quotient

f'(1)		=
		=
		=
		=
		= - (2 + h)
		= - 2

Problem : Give an example of a function that is not differentiable at x = a.

f (x) =|x - a| is not differentiable at x = a because it has a corner there.

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