Exploring the Derivative Rule | Finding the Derivative of a Polynomial with Exponential Term

If f(x)=4x^6−3x^4+2x^3+e^2, then f′(x)=

To find the derivative of a function f(x), we can use the following rules:
1

To find the derivative of a function f(x), we can use the following rules:
1. The derivative of a constant term is 0.
2. The derivative of x^n (where n is a constant) is nx^(n-1).
3. The derivative of e^x is e^x.

In this case, we are given f(x) = 4x^6 – 3x^4 + 2x^3 + e^2. To find f′(x), we need to find the derivative of each term individually and then combine them.

Taking the derivative of 4x^6:
Using rule 2, the derivative of 4x^6 is 6 * 4x^(6-1) = 24x^5.

Taking the derivative of -3x^4:
Using rule 2, the derivative of -3x^4 is 4 * -3x^(4-1) = -12x^3.

Taking the derivative of 2x^3:
Using rule 2, the derivative of 2x^3 is 3 * 2x^(3-1) = 6x^2.

Taking the derivative of e^2:
Using rule 3, the derivative of e^2 is 0.

Now, we can combine these derivatives to find f′(x):
f′(x) = 24x^5 – 12x^3 + 6x^2 + 0
f′(x) = 24x^5 – 12x^3 + 6x^2

Therefore, the derivative of the given function f(x) = 4x^6 – 3x^4 + 2x^3 + e^2 is f′(x) = 24x^5 – 12x^3 + 6x^2.

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