Question

What is the derivative of `y=5^(x-2)`?

Answer 1

`dy/dx= ln5 xx 5^(x- 2)`

Explanation:

Take the natural logarithm of both sides.

`lny = ln(5^(x- 2))`

Now, use the rule `ln(a^n) = nlna`

`lny = (x - 2)ln5`

`d/dx(lny) = d/dx((x - 2)(ln5))`

Differentiating the left-hand side using the rule `d/dx(lnx) = 1/x` and implicit differentiation and the right-hand side using the product rule we have:

`1/y(dy/dx) = 1(ln5) + (x - 2)0`

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

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

`dy/dx= yln5`

`dy/dx= ln5 xx 5^(x- 2)`, since `y = 5^(x- 2)`

Hopefully this helps!