পর্যায়ক্রমিক অন্তরজ (Successive Differentiation)
x=a(t+sint),y=a(1−cost) \mathrm{x}=\mathrm{a}(\mathrm{t}+\sin \mathrm{t}), \mathrm{y}=\mathrm{a}(1-\cos \mathrm{t}) x=a(t+sint),y=a(1−cost) হলে d2ydx2= \frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}= dx2d2y= ?
acost2 \frac{\mathrm{a}}{\operatorname{cost}^{2}} cost2a
1a(1+sint)2 \frac{1}{a(1+\operatorname{sint})^{2}} a(1+sint)21
a(1−cost)2 \frac{\mathrm{a}}{(1-\operatorname{cost})^{2}} (1−cost)2a
costa(1+cost)3 \frac{cost}{a(1+\operatorname{cost})^{3}} a(1+cost)3cost
Solve: dxdt=a(1+cost);dydt=asint \frac{\mathrm{dx}}{\mathrm{dt}}=\mathrm{a}(1+\operatorname{cost}) ; \frac{\mathrm{dy}}{\mathrm{dt}}=\operatorname{asint} dtdx=a(1+cost);dtdy=asint
∴dydx=sint1+cost \therefore \frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\sin \mathrm{t}}{1+\cos \mathrm{t}} ∴dxdy=1+costsintdydx=(1+cost)cost−sint(−sint)(1+cost)2=cost+cos2t+sin2ta(1+cost3 \frac{\mathrm{d}^{\mathrm{y}}}{\mathrm{dx}}=\frac{(1+\operatorname{cost}) \operatorname{cost}-\operatorname{sint}(-\sin t)}{(1+\operatorname{cost})^{2}}=\frac{\operatorname{cost}+\cos ^{2} \mathrm{t}+\sin ^{2} \mathrm{t}}{\mathrm{a}\left(1+\operatorname{cost}^{3}\right.} dxdy=(1+cost)2(1+cost)cost−sint(−sint)=a(1+cost3cost+cos2t+sin2t =costa(1+cost) =\frac{\cos \mathrm{t}}{\mathrm{a}(1+\operatorname{cost})} =a(1+cost)cost
f(x)=lnx,g(x)=(x+1+x2)f(x)=\ln x, g(x)=\left(x+\sqrt{1+x^{2}}\right)f(x)=lnx,g(x)=(x+1+x2)
y=cos2x y=\sqrt{\cos 2 x} y=cos2x হলে, (yy1)2=? \left(y y_{1}\right)^{2}= ? (yy1)2=?
y=1x=x−1 y=\frac{1}{x}=x^{-1} y=x1=x−1 এর n n n তম অন্তরক সহগ নিচের কোনটি ?
y=ex y=e^{x} y=ex হলে, y4 \mathrm{y}_{4} y4 কত ?
