show that d/dx (u^v)=vu^(v-1)du/dx (ln u) (u^v) dv/dx. Hint: let y=u^v and diffe
ID: 3118558 • Letter: S
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show that d/dx (u^v)=vu^(v-1)du/dx (ln u) (u^v) dv/dx. Hint: let y=u^v and differentiate ln y. Use this result to find d/dx (x^x)Explanation / Answer
T.P.==>d(u^v)/dx=vu^(v-1)du/dx (ln u) (u^v) dv/dx take u^v=y ln(y)=vln(u) differentiate both side w.r.t x 1/y * dy/dx = (v/u)du/dx + (ln(u))dv/dx dy/dx = y*(v/u)du/dx + y*(ln(u))dv/dx replace y=u^v dy/dx = (u^v)(v/u)du/dx + (u^v)(ln(u))dv/dx dy.dx=v*u^(v-1) *du/dx + (u^v)(ln(u))dv/dx hence proved now for x^x replace u and v by x in above eqn d(x^x)/dx=x*x(x-1) + (x^x)ln(x) d(x^x)/dx=x^x +(x^x)ln(x) d(x^x)/dx=(x^x)(1+ln(x))
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