Introduction to Derivatives

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

Solution for Problem 1 >>

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

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Problem : f (x) = - x ² . What is the f'(x) ?

Solution for Problem 2 >>

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

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Problem : Find the slope of the line perpendicular to the tangent of the graph of f (x) = - x ² at x = 1.

Solution for Problem 3 >>

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 .

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Problem : Find f'(1) for f (x) = - x ² without solving for f'(x) first.

Solution for Problem 4 >>

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

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