Power rule, f(x)= 2x^3 + 3x^2 + 4x + 5
The power rule is a fundamental concept in calculus that allows you to find the derivative of a function in the form of f(x) = x^n, where n is a constant
The power rule is a fundamental concept in calculus that allows you to find the derivative of a function in the form of f(x) = x^n, where n is a constant.
To apply the power rule, multiply the coefficient of x (in this case, 2) by the exponent (in this case, 3), and then subtract 1 from the exponent. Similarly, do this for each term in the polynomial function.
Applying the power rule to the given function, f(x) = 2x^3 + 3x^2 + 4x + 5, we can find its derivative as follows:
derivative of f(x) = derivative of (2x^3) + derivative of (3x^2) + derivative of (4x) + derivative of (5)
To find the derivative of each term, we apply the power rule as described above:
The derivative of 2x^3 is equal to (2)(3)x^(3-1) = 6x^2
The derivative of 3x^2 is equal to (3)(2)x^(2-1) = 6x
The derivative of 4x is equal to (4)(1)x^(1-1) = 4
The derivative of 5 is equal to 0 (since a constant has a derivative of zero)
Therefore, the derivative of f(x) = 2x^3 + 3x^2 + 4x + 5 is:
f'(x) = 6x^2 + 6x + 4
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