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## AP®︎/College Calculus BC

### Course: AP®︎/College Calculus BC>Unit 2

Lesson 7: Derivative rules: constant, sum, difference, and constant multiple: connecting with the power rule

# Differentiating integer powers (mixed positive and negative)

Let's explore how to differentiate the function g(x) = 2/(x³) - 1/(x²) and evaluate the derivative at x = 2. By applying the Power Rule and simplifying, we efficiently find the slope of the tangent line at the given point.

## Want to join the conversation?

• I don't understand why if we calculate the actual expression then differentiate it the result will be wrong! I mean if we do this: 2x^2 - x^3 / x^5 . then If we differentiate this the result is like 4x - 3x^2 / 5x^4 , if we input 2 here the final result will be -1/20! Why it is like this?
• he has never proved the validity of the power rule for negative powers!
• Do you doubt its validity?
It's not difficult to prove. Start with the differential of x⁻¹ (=1/x)
d/dx (x⁻¹) = Lim (h -> 0) (1/(x+h) - 1/x)/h
= (x - (h+x))/((x+h)xh)
= -h/((x+h)xh) = -1/((x+h)x)
Which as h -> 0 = -1/x²
This agrees with the power rule with n = -1

Now once we have that result we can combine it with the chain rule and the power rule for all other negative powers. eg
x⁻ⁿ = (xⁿ)⁻¹
So d/dx (x⁻ⁿ) = -1 · n·xⁿ⁻¹/(xⁿ)² = -n/x²ⁿ⁻⁽ⁿ⁻¹⁾
= -n/xⁿ⁺¹
Which again agrees with the power rule.
• why does sal leace negative exponents in videos, for example at in this video?
• Well, Sal calculates the negative exponents in the next step. Additionally, he writes the negative exponents out because it can be challenging to calculate them in your head for some.