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

1 Answer
Apr 26, 2017

#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#