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Find the value of dy/dx given that y=a^x |
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Answer» To DIFFERENTIATE a function of the form y=a^x you need to use a neat little trick to rewrite a^x in the form of something you already KNOW how to differentiate. Using the fact that E^ln(x) is equal to x, y = a^x can be written as e^(ln(a)^x) Using log rules ln(a)^x can be written as xlna so now y can now be expressed as y = e^(xlna) This can now be differentiated using the chain rule. Also RECALL that the differential of e^x is e^x. Using these two ideas: where y=e^(xlna) dy/dx = (lna)e^(xlna) now we can substitute in our initial expression y=a^x therefore dy/dx = (a^x)lna. using this method, you can differentiate any function of the same form. for example where y=2^x we can SEE that a=2 so dy/dx = 2^xln2 |
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