Question

If y = 5^x + x^x, Find `(dy)/(dx)`.

Answer 1

y = 5^x + x^x
Let u = 5^x and v = x^x
y = u + v

Differentiating both sides w.r.t.x

`(dy)/(dx) = (du)/(dx) + (dv)/(dx)` 

`(du)/(dx)` = 5^x. log5
Now. v = x^x

Taking log on both sides

log v = log(x^x)
log v= x. log x

`1/v  (dv)/(dx) = x. 1/x` + log x. 1

`therefore (dv)/(dx)` = v [1 + log x]

`therefore (dv)/(dx) = x^x [1 + log x]`

`(dy)/(dx) = (du)/(dx) + (dv)/(dx)` 

`therefore (dy)/(dx)` = 5^x . log 5 + x^x [1 + logx]