Question

Вычислить производную 1. (x*е^х)' 2. (x^3*lnx)' 3. (sin5x)' 4. ((2x-3)^3)' 5. (2x/lnx)'

Answer 1

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