Chain Rule help

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My problem is f(x)=xe^(2+6x^5)

If someone could please tell me how to find the derivative of this problem I would really appreciate it.

 

[mmm4444bot]

I'll transcribe my scribbles.

x * e^(2 + 6x^5) is a product. x is one factor, and the power of e is another factor.

We can use the Product Rule.

Let g(x) = e^(2 + 6x^5)

Then, we see that function f is:

f(x) = x * g(x)

f`(x) = x * g`(x) + (1) * g(x)

We need to calculate g`(x).

g(x) is a composite function. The power of e is the outer function, and the exponent contains the variable x, so the polynomial 6x^5 + 2 is the inner function.

Since the inner function changes the rate at which the power of e changes, we need to use the Chain Rule.

The power of e is itself a function of x

I see it as e^2 * e^(6x^5).

I mean, it is a function of the form g(n) = C * e^n where C is a constant (e^2) and n is the independent variable (6x^5).

g`(n) = C * e^n * ln(e)

Therefore, we have the following.

g(x) = e^2 * e^(6x^5)

g`(x) = e^2 * e^(6x^5) * ln(e) * 30x^4

or

g'(x) = 30x^4 * e^(2 + 6x^5) * ln(e)

Can you put this together, into the product rule in blue (above) ?