# use a chain rule to find the derivative of f(x)=e^3x

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Given f(x) = e^3x

We need to find the first derivative f'(x) using the chain rule.

We will assume that u= 3x ==> u' = 3

==> f(x) = e^u

==> Now we will differentiate:

==> f(x) = (e^u)' = u' * e^u du

==> Now we will substitute with x.

==> f(x) = 3 ** e^3x = 3*e^3x

**Thenthe derivative of f(x) is f'(x) = 3*e^3x**

We have to find the chain rule to find the derivative of f(x) = e^3x.

The chain rule for the function f(x) = g(h(x)) gives f'(x) = g'(h(x))*h'(x).

Here f(x) = e^3x

f'(x) = e^3x * (3x)'

=> e^3x * 3

**The required derivative is 3*e^3x.**

Since the given function is a composed function, we'll apply chain rule to find it's derivative:

We'll have (u(v(x)) = e^3x

u(v) = e^v => u'(v) = e^v (u is differentiated with respect to v)

v(x) = 3x => v'(x) = 3 (v is differentiated with respect to x).

f'(x) = (u(v(x))' = v'(x)*e^v

f'(x) = 3*e^3x

**The result of differentiation of the given function is: f'(x) = 3e^3x.**

f(x)=e^3x

Remember that the derivative of e^u is the derivative of u times e^u

so:

f'(x) = 3e^3x