Question

find the value of dy/dx
given that y=a^x​

Answer 1

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