Evaluate each of the following. Show all your calculations.
f'(���3)� if f(x) ���� = x^4�� − 3x�
To evaluate the derivative of the function f(x) = x^4 - 3x at x = 3, we need to find f'(3), which represents the value of the derivative at x = 3.
The derivative of a function can be found using the power rule:
If f(x) = x^n, then f'(x) = nx^(n-1)
In this case, f(x) = x^4 - 3x. Applying the power rule, we have:
f'(x) = d/dx (x^4) - d/dx (3x)
= 4x^(4-1) - 3
= 4x^3 - 3
To evaluate f'(3), we substitute x = 3 into the derivative expression:
f'(3) = 4(3^3) - 3
= 4(27) - 3
= 108 - 3
= 105
Therefore, f'(3) = 105.