Question

Вычислить производную функции.

Answer 1

$1) f(x)=x^{10}+x^8+x^3,\f'(x)=left(x^{10}+x^8+x^3right)'=left(x^{10}right)'+left(x^8right)'+left(x^3right)'=\=10x^{10-1}+8x^{8-1}+3x^{3-1}=10x^9+8x^7+3x^2;\\2) f(x)=7x^5-3x^9+4,\f'(x)=left(7x^5-3x^9+4right)'=left(7x^5right)'-left(3x^9right)'+4'=\=7left(x^5right)'-3left(x^9right)'+0=7cdot4x^{5-1}-3cdot9x^{9-1}=28x^4-27x^8;\\3) f(x)=4sqrt x+frac{x}{3}+frac{5}{x},$
$f'(x)=left(4sqrt x+frac{x}{3}+frac{5}{x}right)'=left(4sqrt {x}right)'+left(frac{x}{3}right)'+left(frac{5}{x}right)'=\\=4left(x^{frac{1}{2}}right)'+frac{x'cdot3-xcdot3'}{3^2}+frac{5'x-5x'}{x^2}=4cdotfrac{1}{2}cdot x^{frac{1}{2}-1}+frac{1cdot3-xcdot0}{9}+frac{0cdot x-5cdot1}{x^2}=\\=2x^{-frac{1}{2}}+frac{3}{9}+frac{-5}{x^2}=frac{2}{sqrt{x}}+frac{1}{3}-frac{5}{x^2};$

$4) f(x)=frac{4}{x}-frac{sqrt x}{4}=4cdotfrac{1}{x}-frac{1}{4}cdotsqrt{x}=4x^{-1}-frac{1}{4}cdotsqrt{x},\\f'(x)=left(4x^{-1}-frac{1}{4}cdotsqrt{x}right)'=left(4x^{-1}right)'-left(frac{1}{4}cdotsqrt{x}right)'=\\=4left(x^{-1}right)'-frac{1}{4}left(sqrt{x}right)'=4cdot(-1)x^{-1-1}-frac{1}{4}left(x^{frac{1}{2}}right)'=\\=-4x^{-2}-frac{1}{4}cdotfrac{1}{2}x^{frac{1}{2}-1}=-frac{4}{x^2}-frac{1}{8}x^{-frac{1}{2}}=-frac{4}{x^2}-frac{1}{8sqrt{x}}.$