Constant Multiple & Sum Rules
Two simple but essential rules: how to handle constants and how to handle sums.
Constant Multiple Rule
d/dx[c*f(x)] = c*f'(x)
Constants factor out of derivatives
Sum/Difference Rule
d/dx[f(x) +/- g(x)] = f'(x) +/- g'(x)
Derivative of a sum is the sum of derivatives
Polynomial Derivative
Easy
Find d/dx[3x^4 - 5x^2 + 7x - 9]
1
Apply rules term by term
= d/dx[3x^4] - d/dx[5x^2] + d/dx[7x] - d/dx[9]
2
Use power and constant rules
= 3(4x^3) - 5(2x) + 7(1) - 0
3
Simplify
= 12x^3 - 10x + 7
Answer:
12x^3 - 10x + 7
Practice
1. d/dx[5x^3 + 2x - 1] = ?
Key Takeaways
Constants factor out: d/dx[cf] = cf'
Take derivatives term by term
Constant derivative is 0: d/dx[c] = 0