Use the limit definition of the derivative to find 𝑓′(𝑥) if 𝑓(𝑥) = 𝑥 10?

Question

Use the limit definition of the derivative to find 𝑓′(𝑥) if 𝑓(𝑥) = 𝑥
10. Expand all the way out. I came up with a 10 term answer, not sure if that correct or not.
Thank you!

Ash · Accepted Answer

𝑓(𝑥) = 𝑥¹⁰
𝑓'(𝑥) = lim(h → 0) [𝑓(𝑥+h) - 𝑓(𝑥)]/h
𝑓'(𝑥) = lim(h → 0) [(𝑥+h)¹⁰ - 𝑥¹⁰]/h
Using Pascal's triangle expand (𝑥+h)¹⁰
𝑓'(𝑥) = lim(h → 0) [𝑥¹⁰ + 10𝑥⁹h + 45𝑥⁸h² + 120𝑥⁷h³ + 210𝑥⁶h⁴ + 252𝑥⁵h⁵ + 210𝑥⁴h⁶ + 120𝑥³h⁷ + 45𝑥²h⁸ + 10𝑥h⁹ + h¹⁰ - 𝑥¹⁰]/h
𝑓'(𝑥) = lim(h → 0) [10𝑥⁹h + 45𝑥⁸h² + 120𝑥⁷h³ + 210𝑥⁶h⁴ + 252𝑥⁵h⁵ + 210𝑥⁴h⁶ + 120𝑥³h⁷ + 45𝑥²h⁸ + 10𝑥h⁹ + h¹⁰]/h
𝑓'(𝑥) = lim(h → 0) h[10𝑥⁹ + 45𝑥⁸h + 120𝑥⁷h² + 210𝑥⁶h³ + 252𝑥⁵h⁴ + 210𝑥⁴h⁵ + 120𝑥³h⁶ + 45𝑥²h⁷ + 10𝑥h⁸ + h⁹]/h
𝑓'(𝑥) = lim(h → 0) [10𝑥⁹ + 45𝑥⁸h + 120𝑥⁷h² + 210𝑥⁶h³ + 252𝑥⁵h⁴ + 210𝑥⁴h⁵ + 120𝑥³h⁶ + 45𝑥²h⁷ + 10𝑥h⁸ + h⁹]

Now plug in h=0
𝑓'(𝑥) = 10𝑥⁹

az_lender · Answer

No, you don't need 10 terms.

f(x+h) = (x+h)^10.

f(x+h) - f(x) = 10*x^9*h + 45*x^8*h^2 + 120*x^7*h^3 + blah blah blah + h^10.

When you divide this by h, you get

[f(x+h) - f(x)]/h = 10*x^9 + 45*x^8*h + 120*x^7*h^2 + blah blah blah + h^9.

Now, when you take the limit as h-->0, you get only 10x^9.  Period!

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Use the limit definition of the derivative to find 𝑓′(𝑥) if 𝑓(𝑥) = 𝑥 10?

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