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Related Rates Problem
Problem 163: A growing sand pile Sand falls from a conveyor belt at the rate of 10 onto the top of a conical pile. The height of the pile is always three-eighths of the base diameter. How fast are the (a) height and (b) radius changing when the pile is 4 m high? Answer in…
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Derivative of the Logarithmic Function for Arbitrary Base
Problem 162: Show that , where is a differentiable function and is an arbitrary base. Solution: Using the relationship between and , (1) Then (2) That is, (3)
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Derivative of A Constant Raised to a Power
Problem 161: Show that where and is a differentiable function of . Solution: Using the relationship between and , (1) Then (2) That is, (3)
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Logarithmic Differentiation
Problem 160: Find the derivative of (1) Solution: We will use properties of logarithmic to solve this problem. That is, (2)
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Mathematical Induction
Problem 159: Using mathematical induction, show that for (1) Solution: Since we will use mathematical induction to prove (1), we need to prove the base case and the inductive case. That is, Base Case: Let . Then (2) Inductive Case: We assume that (1) holds for . We will show that it also…
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Derivative of Inverse CSC
Problem 158: Use the Identity (1) to derive the formula for the derivative of . Solution: Using the identity (1), we have (2) That is, (3)
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The Number e as a Limit
Problem 157: Show that the number can be calculated as the limit (1) Solution: Let . Then at . But using the definition of derivative, (2) Because and is given by (2), they must be equal. That is, (3) That is, (4)
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Derivative of Composite Function
Problem 156: Compute the derivative of . Solution: We need to recall that if and , then (1) This is nothing more than the derivative of the composition of function (aka the chain rule). That is, in this case, we have so that . This means that (2) Hence,
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Differentiability at a Point
Problem 155: Show that the function is differentiable on and on but has no derivative at . Solution: A function is differentiable if the function has a derivative. That is, (1) On , . Then . Therefore, on , the function has derivative -1. (2) On , . Then . Therefore, on , the function…
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Derivative as a Rate of Change II
Problem 154: It takes 12 hours to drain a storage tank by opening the valve at the bottom. The depth of fluid in the tank hours after the valve is opened is given by the formula (1) a. Find the rate (m/h) at which the tank is draining at time . b. When is…
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