Question

Question #3b716

Answer 1

I assume that the poster meant: `y=e^x tan sqrt{5x}`

By rewriting the square-root as the 1/2-power,

`y=e^xtan(5x)^{1/2}`

Let us now find the derivative.

By Product Rule,

`y'=(e^x)'cdot tan(5x)^{1/2}+e^x cdot( tan(5x)^{1/2})'`

by Exponential Rule and Chain Rule,

`=e^xtan(5x)^{1/2}+e^x sec^2(5x)^{1/2}cdot[(5x)^{1/2}]'`

by Chain Rule,

`=e^xtan(5x)^{1/2}+e^x sec^2(5x)^{1/2}cdot1/2(5x)^{-1/2}cdot5`

by cleaning up a bit,

`=e^x(tan sqrt{5x}+{5sec^2 sqrt{5x}}/{2sqrt{5x}})`

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I hope that this was helpful.