Question

Вычислить производную

1. (X*е^х)'
2. (X^3*lnx)'
3. (Sin5x)'
4. ((2x-3)^3)'
5. (2x/lnx)'

Answer 1

$(xcdot e^{x})'=e^{x}+xcdot e^{x}=e^{x}(1+x)\\(x^3cdot lnx)'=3x^2cdot lnx+x^3cdot frac{1}{x}=x^2cdot (3lnx+1)\\(sin5x)'=cos5xcdot underbrace {(5x)'}_{5}=5cos5x\\((2x-3)^3)'=3(2x-3)^2cdot underbrace{(2x-3)'}_{2}=6(2x-3)^2\\(frac{2x}{lnx})'=frac{2cdot lnx-2xcdot frac{1}{x}}{ln^2x}=frac{2(lnx-1)}{ln^2x}$