Question

a₀ xⁿ + a₁ xⁿ⁻¹ + a₂ xⁿ⁻² + ... + a_n−1 x + a_n. 

Answer 1

\[\frac{d}{dx}\left[ \left( a_0 x^n \right) + \frac{d}{dx}\left( a_1 x^{n - 1} \right) + \frac{d}{dx}\left( a_2 x^{n - 2} \right) + . . . + a_{n - 1} x + a_n \right]\]
\[ = a_0 \frac{d}{dx}\left( x^n \right) + a_1 \frac{d}{dx}\left( x^{n - 1} \right) + a_2 \frac{d}{dx}\left( x^{n - 2} \right) + . . . + a_{n - 1} \frac{d}{dx}\left( x \right) + \frac{d}{dx}\left( a_n \right)\]
\[ = n a_0 x^{n - 1} + \left( n - 1 \right) a_1 x^{n - 2} + \left( n - 2 \right) a_2 x^{n - 3} + . . . . + a_{n - 1} \left( 1 \right) + 0\]
\[ = n a_0 x^{n - 1} + \left( n - 1 \right) a_1 x^{n - 2} + \left( n - 2 \right) a_2 x^{n - 3} + . . . . + a_{n - 1} \]
\[\]