Problem 24 Medium Difficulty

Prove the formulas given in the Table 1 for the derivatives of the functions (a) $ \cosh, $ (b) $ \tanh, $ (c) csch, (d) sech, and (e) $ \coth. $

Answer

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it's clear. So when you married here So we have d of clash of that's over. D s is equal to 1/2 de eat the X Let's eat to the negative x over d x This becomes equal to one huh of e to the X minus e to the negative x so they see that D of D over DX of caution is equal to the high public function of sign. For part B, we have the 10 which is equal to sign over. It's not co sign when we get e to the X miners, Pete the negative x all over e to the X plus e to the negative x So we're gonna differentiate you want over d X We eat to the X minus B to the negative x the derivative beach the X plus e to the negative x minus e to the X minus each. The negative bets That's a derivative of e to the X plus eat. Then they go all over each the X plus e to the negative explorer. We end up getting e to the two x plus two plus eats. The negative to X was too minus eats in the negative to us all over b to the X Let's eat the negative X square, which is equal to two over e to the X plus e to the negative X square. Then we get one over you two x plus e to the negative X over two square. We're just equal to one over the high public function of CO sign square when we get sequined square. No, that's report. See, we have Coast seeking is equal to one over sign which is equal to one over when, huh e to the X minus e to the negative effects, which is equal to two terms. Eat the explain. It eats the negative eggs, the negative one power we differentiate. Do you? Why? Over d x we get negative two times negative one terms Eat the X minus e to the negative effects to the negative sucking power times The derivative of the two x minus e. The negative butts, which is equal to negative two over e to the X minus, eats the negative. X comes eats the X plus e to the negative effects all over e to the X minus negative each. That's negative. It's which is equal to negative. Cosi Comte Times Co Sign over Sign This is equal to negative coast. He can't times put engines for party. We're gonna do another proof. So we get second is equal to one over co sign jizzy bolted to beat the X plus e to the negative extra negative first power we differentiate and then we get negative one times two each the X plus e to the negative extra second power times eat the eggs plus e to the negative x the derivative. This becomes equal to negative too well over e to the X plus e to the negative X worms Eat the oneness e to the negative effects all over e to the X plus eats the negative x the second port. It becomes 1/2 e to the x minus e to the negative x over 1/2 b to the X plus each the negative x Jake's equal to negative The hyperbole function of secrets I'm sign over co sign just equal to the negative sequence and the hiker of like function of tangent her e we're looking at cu attention. Is it equal to each the explosive e to the negative X all over each the X minus e to the negative effects when we get the derivative equal to eat to the X plus each the negative driven it off that pizza X minus E to the negative effects minus eat X plus eats and negative X terms each the X minus each than negative X and the derivative function that all over e to the x minus e to the negative x where which becomes equal to negative to over eat the X minus. Eat the negative X square, which is equal to one over negative one over. I probably function of sign when we got co Siegen Negative Cosi Graham Other X squared.