Multiply by. Reorder and. The final answer is. Plug in the components.
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Simplify the numerator. Factor out of.
Divide by. Take the limit of each term. Split the limit using the Sum of Limits Rule on the limit as approaches. Split the limit using the Product of Limits Rule on the limit as approaches.
Move the term outside of the limit because it is constant with respect to. Evaluate the limits by plugging in for all occurrences of. Evaluate the limit of which is constant as approaches. Evaluate the limit of by plugging in for.
Simplify the answer. The slope is and the point is.
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Find the value of using the formula for the equation of a line. Use the formula for the equation of a line to find. Substitute the value of into the equation. Find the value of. We call that limit the function f ' x -- " f -prime of x " -- and when that limit exists, we say that f itself is differentiable at x , and that f has a derivative. And so we take the limit of the difference quotient as h approaches 0. When that limit exists, that means that the difference quotient can be made as close to that limit -- " f ' x " -- as we please. As for x , we are to regard it as fixed.
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In practice, we have to simplify the difference quotient before letting h approach 0. We have to express the numerator To sum up: The derivative is a function -- a rule -- that assigns to each value of x the slope of the tangent line at the point x , f x on the graph of f x.
It is the rate of change of f x at that point. Here is the difference quotient , which we will proceed to simplify:. Lesson 18 of Algebra. In going to line 3 , we subtracted the x 2 s. That is, we subtracted f x. In going to line 4 , we divided the numerator by h. Lesson 20 of Algebra.
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We can do that because h is never equal to 0, even when we take the limit Lesson 2. Whenever we apply the definition, we have to algebraically manipulate the difference quotient so that we can simply replace h with 0. In fact, the entire theory of limits, with all its complexities and subtleties, was invented to justify just that. Poor Newton and Leibniz were criticized for offering justifications that the 19th century inventors of limits didn't like. According to the definition, a function will be differentiable at x if a certain limit exists there.
Graphically, this means that the graph at that value of x will have a tangent line. At which values, then, would a function not be differentiable? Where it does not have a tangent line. Above are two examples. Topic 5 of Precalculus. For, the left-hand limit of the function itself as x approaches 0 is equal to the right-hand limit, namely 0.
This illustrates that continuity at a point is no guarantee of differentiability -- the existence of a tangent -- at that point. Conversely, though, if a function is differentiable at a point -- if there is a tangent -- it will also be continuous there. The graph will be smooth and have no break. Since differential calculus is the study of derivatives, it is fundamentally concerned with functions that are differentiable at all values of their domains.
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To cover the answer again, click "Refresh" "Reload". Think about this yourself first! We are to take the derivative of what follows it.