Constant Multiple & Sum Rules

Business Calculus · Axiom Academy

LESSON Constant Multiple & Sum Rules Differentiating functions term by term Before diving into combinations, let's cover the simplest derivative rule: The derivative of a constant is zero. Why? A constant function like f(x) = 5 is a horizontal line. Since it's flat, its slope is zero everywhere! Constants don't change, so their rate of change is zero. Examples: the derivative of 7 is 0, the derivative of -100 is 0, the derivative of is 0. What happens when we multiply a function by a constant? The derivative of a constant times a function equals the constant times the derivative of the function. "Constants can be pulled out of derivatives" In practice, you can do this in one step: bring down the exponent and multiply by the existing coefficient! How do we differentiate a sum or difference of functions? Derivative of a sum = sum of derivatives Derivative of a difference = difference of derivatives You can differentiate term by term ! This is incredibly powerful because it lets us find derivatives of polynomials by handling each term separately. Example: Find the derivative of f(x) = 3x^4 - 5x^2 + 7x - 2 Click each card to reveal the derivative. Try to solve it first! Don't confuse these rules with product/quotient rules! The sum rule works for f + g , NOT for f g or g . We'll cover those next! What is the derivative of f(x) = 6x^4 - 2x^3 + 5x - 8 ? You can now differentiate any polynomial! The key rules:

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