f(x)= e^-3x
g(x)=x+x^2
calculate the following derivative
(d/dx)f(g(x))=?
To calculate the derivative of the composition of two functions f(g(x)), we can use the chain rule.
The chain rule states that if we have a composition of two functions, u(x) = f(g(x)), then the derivative of u(x) with respect to x is given by:
du/dx = du/dg * dg/dx
Let's calculate the derivative step by step.
First, let's find the derivative of f(g(x)) with respect to g(x) (du/dg). Since f(x) = e^(-3x), the derivative of f(g(x)) with respect to g(x) can be calculated as:
df(g(x))/dg = d/dg (e^(-3g(x)))
To find this derivative, we use the chain rule again, as we have a composition within a composition. The derivative of e^(-3g(x)) with respect to g(x) can be calculated as:
d/dg (e^(-3g(x))) = (-3) * e^(-3g(x))
Now, let's find the derivative of g(x) with respect to x (dg/dx). Since g(x) = x + x^2, the derivative of g(x) with respect to x can be calculated as:
dg(x)/dx = d/dx (x + x^2)
Taking the derivative of each term separately, the derivative of x with respect to x is simply 1, and the derivative of x^2 with respect to x can be calculated using the power rule, which states that the derivative of x^n with respect to x is n*x^(n-1). Applying this rule, we get:
dg(x)/dx = 1 + 2x
Now, we can finally calculate the derivative of f(g(x)) with respect to x (du/dx) using the chain rule:
du/dx = du/dg * dg/dx
Substituting the values we found for du/dg and dg/dx, we get:
du/dx = (-3) * e^(-3g(x)) * (1 + 2x)
So, the derivative of f(g(x)) with respect to x is (-3) * e^(-3g(x)) * (1 + 2x).