What is the first derivative of the following function: f(x) = a^(-bx)?
(I am lost with this question)
Review your chain rule: if
y = a^u where u is a function of x, then
y' = ln(a) a^u u'
So, for this one,
f'(x) = ln(a) a^(-bx) (-b)
To find the first derivative of the function f(x) = a^(-bx), we can use the power rule of differentiation. The power rule states that if we have a function of the form g(x) = x^n, then the derivative of g(x) (denoted as g'(x)) is given by g'(x) = nx^(n-1).
In our case, we have f(x) = a^(-bx), where a and b are constants. To apply the power rule, we first rewrite the function as:
f(x) = (a^(-b))^x
Now we can see that the function has the form g(x) = u^x, where u = a^(-b). Applying the power rule, we have:
f'(x) = ln(u) * u^x
Therefore, to find the first derivative of f(x) = a^(-bx), you can calculate ln(a^(-b)) and multiply it by a^(-bx).