# Nth Derivative of a Polynomial (part 1)

In calculus, the derivative of a polynomial is:$${ ax }^{ n }={ nax }^{ n-1 }$$. But how could we know the 2nd, 3rd or even the 100th derivative of a given polynomial expression. I formulated new formulas that would ensure that the time in finding the number of derivative would be less in calculating the higher derivatives. These are the following conditions:

Let's assume that n is the no. of differentiation or derivatives that a polynomial must derive and b be the exponents of each term in a given polynomial.

1. If $$n<b$$ , then the nth derivative of the polynomial would be

$$\frac { { d }^{ n } }{ { dx }^{ n } } { ax }^{ b }=a(b-n+1)(b-n+2)....(b-n+n){ x }^{ b-n }$$.

Example: Find the third derivative of $${ 4x }^{ 5 }$$?

Solution: Since the condition is $$n<b$$ then let's use the formula stated in first condition. Let n be 3 and b be 5.Therefore, the solution would be:

$$\frac { { d }^{ 3 } }{ { dx }^{ 3 } } { 4x }^{ 5 }=4(5-3+1)(5-3+2)(5-3+3){ x }^{ 5-3 }$$

$$\frac { { d }^{ 3 } }{ { dx }^{ 3 } } { 4x }^{ 5 }=4(3)(4)(5){ x }^{ 5-3 }$$

$$\frac { { d }^{ 3 } }{ { dx }^{ 3 } } { 4x }^{ 5 }={ 240x }^{ 2 }$$

Note by Merzel Mark Guilaran
3 years, 9 months ago

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Yes in general $$\dfrac{d^n}{dx^n} (ax^r) = an! {\dbinom{r}{n}} x^{r-n}$$. Good insight by the way sir.

- 6 months, 2 weeks ago