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Math 180. Packet #10 Proving Derivatives. Proof of Power Rule Suppose . Then,. Proof of Power Rule Suppose . Then,. Proof of Power Rule Suppose . Then,. Proof of Power Rule Suppose . Then,. Proof of Power Rule Suppose . Then,. Proof of Power Rule Suppose . Then,.
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Math 180 Packet #10 Proving Derivatives
Proof of Power Rule Suppose . Then,
Proof of Power Rule Suppose . Then,
Proof of Power Rule Suppose . Then,
Proof of Power Rule Suppose . Then,
Proof of Power Rule Suppose . Then,
Proof of Power Rule Suppose . Then,
Proof of Power Rule Suppose . Then,
Trigonometric Functions To figure out the derivatives of and by the definition, we need these two limits: and These two limits can be proven using a geometric argument. Then we can find the derivatives of and as follows:
Trigonometric Functions To figure out the derivatives of and by the definition, we need these two limits: and These two limits can be proven using a geometric argument. Then we can find the derivatives of and as follows:
Trigonometric Functions But what about and the rest of the gang? Ex 1.Prove that the derivative of is by using the derivatives of and .
Exponential Functions To find the derivative of by the definition, we need the following limit: This limit is sometimes taken to just be definitional. That is, is the number where .
Exponential Functions To find the derivative of by the definition, we need the following limit: This limit is sometimes taken to just be definitional. That is, is the number where .
Exponential Functions Then,
Exponential Functions Then,
Exponential Functions Then,
Exponential Functions Then,
Exponential Functions Then,
Exponential Functions Here’s how to get the derivative of :
Exponential Functions Here’s how to get the derivative of :
Exponential Functions Here’s how to get the derivative of :
Exponential Functions Here’s how to get the derivative of :
Exponential Functions Here’s how to get the derivative of :
Logarithmic Functions To find the derivative of logarithms, we can use implicit differentiation. Ex 2.Prove that the derivative of is .
Logarithmic Functions Note: We can extend our results to include negative -values:
Logarithmic Functions Here’s how to get the derivative of .
Logarithmic Functions Here’s how to get the derivative of .
Logarithmic Functions Here’s how to get the derivative of .
Logarithmic Functions Here’s how to get the derivative of .
Logarithmic Functions Here’s how to get the derivative of .
Inverse Trigonometric Functions We can use implicit differentiation with inverse trig functions, too. Ex 3.Prove that the derivative of is .
Hyperbolic Functions Recall that and . Ex 4.Prove that the derivative of is .