Question

Find `"dy"/"dx"` if y = x^x + 5^x

Answer 1

y = x^x + 5^x 

Let u = x^x  , v = 5^x  

Taking log on both sides of u 

log u = x log x 

Dtffercnttallng w.r.t. x 

`therefore 1/"u" "du"/"dx" = "x" (1/"x") + "log x" . (1)`

`therefore "du"/"dx" = "u" (1 + "log x")`

= x^x (1 + log x)

Differentiating v. w.r.t. x 

`"dv"/"dx"` = 5^x log 5

As y = u + v

`therefore "dy"/"dx" = "du"/"dx" + "dv"/"dx"`

`therefore "dy"/"dx" = "x"^"x" (1 + "log x") + 5^"x" "log" 5`