Question

How do I find `f'(x)` for `f(x)=5^x` ?

Answer 1

`dy/dx = 5^x log5`

Explanation:

`y=5^x`

`logy=x log5`

`dy/dx. 1/y=x.(0)+log5.(1)`

`dy/dx . 1/y =log5`

`dy/dx =log5 . y`

`dy/dx =log5 . 5^x`

`dy/dx = 5^x log5`

Answer 2

`f(x)=y=5^x`

Taking log on both the sides,

`logy=log5^x`

`logy=xlog5` `color(white)(wwggggggggggggw` `["as " color(red)(loga^b = bloga)]`

Now, applying implicit differentiation,
`1/y dy/dx=x (d(log5))/dx + log5` `color(white)(o` `["product rule is applied on "xlog5 ]`

`dy/dx=y[x + log5]` `color(white)(gggggggw` `["as derivative of a constant is 1" ]`

`dy/dx=5^x[x + log5]`

Answer 3

`f'(x)=ln5*5^x`

Explanation:

Let `y=f(x)=5^x`

then `lny=xln5` and differentiating we get

`1/y(dy)/(dx)=ln5`

or `(dy)/(dx)=ln5*y=ln5*5^x`

Answer 4

`5^x*ln5`

Explanation:

Given: `f(x)=5^x`

Using the common integral that `(a^x)'=a^xlna`, we get:

`f'(x)=5^x*ln5`