Question

How do you differentiate `y =5^x log_5 x`?

Answer 1

`dy/dx=5^x/(xln5)+5^xln5log_5x`

Explanation:

differentiate using the `color(blue)"product rule"`

`"Given " y=g(x)h(x)" then "`

`dy/dx=g(x)h'(x)+h(x)g'(x)larr" product rule"`

`"using the following "color(blue)"standard derivatives"`

`• d/dx(a^x)=a^xlna`

`• d/dx(log_ax)=1/(xlna)`

`"here "g(x)=5^xrArrg'(x)=5^xln5`

`"and " h(x)=log_5xrArrh'(x)=1/(xln5)`

`rArrdy/dx=5^x. 1/(xln5)+log_5x.5^xln5`

`color(white)(rArrdy/dx)=5^x/(xln5)+5^xln5log_5x`