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Find the derivative of $f(x) = 3x^2 + 5x - 2$.
Apply the sum/difference rule and constant multiple rule: $f'(x) = 6x + 5$.
Find the derivative of $f(x) = 7$.
Apply the constant rule: $f'(x) = 0$.
Find the derivative of $f(x) = 4x^3 - 2x + 1$.
Apply the sum/difference rule and constant multiple rule: $f'(x) = 12x^2 - 2$.
Find the derivative of $f(x) = 5x^4 + 3$.
Apply the sum rule, constant rule, and constant multiple rule: $f'(x) = 20x^3$.
Find the derivative of $f(x) = 2x^2 - 6x$.
Apply the difference rule and constant multiple rule: $f'(x) = 4x - 6$.
Explain the Constant Rule.
The derivative of a constant is always zero because a constant's value does not change.
Explain the Sum Rule.
When differentiating a sum of functions, you can differentiate each function separately and then add the results.
Explain the Difference Rule.
When differentiating a difference of functions, you can differentiate each function separately and then subtract the results.
Explain the Constant Multiple Rule.
A constant multiplying a function remains as a multiplier when you take the derivative of the function.
Constant Rule Formula
If $f(x) = c$, then $f'(x) = 0$.
Sum Rule Formula
If $f(x) = g(x) + h(x)$, then $f'(x) = g'(x) + h'(x)$.
Difference Rule Formula
If $f(x) = g(x) - h(x)$, then $f'(x) = g'(x) - h'(x)$.
Constant Multiple Rule Formula
If $f(x) = c cdot g(x)$, then $f'(x) = c cdot g'(x)$.