`f(x) =x9^x` Find the derivative of the function

We can apply the Product Rule for derivatives:

`d/(dx)(u*v) = u' *v + u * v` '.

With the given function:` f(x) =x*9^x` , we may let:

`u = x` then `u ' = 1`

Using the formula for the derivative of an exponential function:

`d/(dx)(a^u) =a^u* ln(a)*(du)/(dx) ` where ` a!=1` .

Then `d/(dx)(9^x) = 9^x * ln(9) = 9^xln(9)` where ` a=9 ` and` u =x` .

Using Product Rule with the values:

`u=x` , `u'= 1` ,`v=9^x ` and v`'= 9^xln(9)` ,we get:

`f'(x)= d/(dx) (x* 9^x)= 1 * 9^x + x* 9^xln(9)`

`f'(x) =9^x +x9^xln(9)`

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