Sauter

Section 2.1 Règles de dérivation

Calculez \(\displaystyle\frac{dy}{dx}\text{.}\)

  1. \(\displaystyle y=\left(x^3+x^5\right)^7\)

  2. \(\displaystyle y=\frac{3}{\sqrt[3]{x}}-\frac{7}{\sqrt[7]{x^5}}\)

  3. \(\displaystyle y=\frac{x^2+3x^4-5}{\sqrt{x}}\)

  4. \(\displaystyle y=\frac{\cos(x)}{1+\sin(x)}\)

  5. \(\displaystyle y=x\arcsin(x)\)

  6. \(\displaystyle xe^y=y\sin(x)\)

  7. \(\displaystyle y=\frac{s^5+e^t}{t^2+t+1}\)

  8. \(\displaystyle y=x\ln\left(x\ln(x)\right)\)

  9. \(\displaystyle y=\sqrt{x}\tan\left(\sqrt{x}\right)\)

  10. \(\displaystyle y=\sqrt{\arctan(x)}\)

  11. \(\displaystyle y=x^3\arcsin(2x+3)\)

  12. \(\displaystyle y=e^{\sin(\pi x)}\)

  13. \(\displaystyle y=\sqrt{\ln\left(x^2\right)}\)

  14. \(\displaystyle y=7^{\cos(x)}\)

  15. \(\displaystyle y=3^{\sin(x^2)}\)

  16. \(\displaystyle \cos(xy)=y-x^2\)

  17. \(\displaystyle y=\log_3(3x+2)\)

  18. \(\displaystyle y=(\sin(x))^x\)

  19. \(\displaystyle y+x\cos(y)=x^2y\)

  20. \(\displaystyle y=\ln\left(\frac{x^2-9}{x^3+27}\right)\)

Réponse

  1. \(\displaystyle y'=7\left(x^3+x^5\right)^6(3x^2+5x^4)\)

  2. \(\displaystyle y'=\frac{5}{\sqrt[7]{x^{12}}}-\frac{1}{\sqrt[3]{x^4}}\)

  3. \(\displaystyle y'=\frac{3\sqrt{x}}{2}+\frac{21\sqrt{x^5}}{2}+\frac{5}{2\sqrt{x^3}}\)

  4. \(\displaystyle y'=-\frac{1}{1+\sin(x)}\)

  5. \(\displaystyle y'=\arcsin(x)+\frac{x}{\sqrt{1-x^2}}\)

  6. \(\displaystyle y'=\frac{y\cos(x)-e^y}{xe^y-\sin(x)}\)

  7. \(\displaystyle y'=0\)

  8. \(\displaystyle y'=\ln\left(x\ln(x)\right)+1+\frac{1}{\ln(x)}\)

  9. \(\displaystyle y'=\frac{\tan\left(\sqrt{x}\right)}{2\sqrt{x}}+\frac{\sec^2\left(\sqrt{x}\right)}{2}\)

  10. \(\displaystyle y'=\frac{1}{2(1+x^2)\sqrt{\arctan(x)}}\)

  11. \(\displaystyle y'=3x^2\arcsin(2x+3)+\frac{2x^3}{\sqrt{1-(2x+3)^2}}\)

  12. \(\displaystyle y'=\pi\cos(\pi x)e^{\sin(\pi x)}\)

  13. \(\displaystyle y'=\frac{1}{x\sqrt{\ln\left(x^2\right)}}\)

  14. \(\displaystyle y'=-\ln(7)\sin(x)7^{\cos(x)}\)

  15. \(\displaystyle y'=2\ln(3)x\cos(x^2)3^{\sin(x^2)}\)

  16. \(\displaystyle y'=\frac{2x-\sin(xy)}{1+x\sin(xy)}\)

  17. \(\displaystyle y'=\frac{3}{\ln(3)(3x+2)}\)

  18. \(\displaystyle y'=\left(\ln(\sin(x))+x\cot(x)\right)(\sin(x))^x\)

  19. \(\displaystyle y'=\frac{2xy-\cos(y)}{1-x^2-x\sin(y)}\)

  20. \(\displaystyle y'=\frac{2x}{x^2-9}-\frac{3x^2}{x^3+27}\)

Évaluez :

  1. \(\displaystyle f'(x)\) si \(\displaystyle f(x)=3x^{\frac{1}{3}}+\frac{7}{4x^{\frac{3}{4}}}-\frac{2x^{\frac{5}{2}}}{5}+4^4\)

  2. \(\displaystyle f'(1)\) si \(\displaystyle f(x)=x^4(x^3-\sqrt{x})+\sqrt{3x}-3\sqrt{x}-\sqrt{\frac{x}{3}}\)

  3. \(\displaystyle \frac{dy}{dx}\) si \(\displaystyle y=(x^2+3)^4(2x^3-5)^3\)

  4. \(\displaystyle \frac{dy}{dx}\) si \(\displaystyle y=\left(\frac{x}{7+x}\right)^5\)

  5. \(\displaystyle \frac{dh}{dt}\) si \(\displaystyle h(t)=\sqrt[3]{(1-t)^3+\sqrt{t^2-2t}}\)

  6. \(\displaystyle \left.\frac{dg}{dt}\right|_{t=-2}\) si \(\displaystyle g(t)=\frac{t}{t+1}+\frac{t+1}{t^2}\)

  7. \(\displaystyle \left.\frac{dy}{dz}\right|_{z=1/2}\) si \(\displaystyle y=5x^2-\sqrt{x}\quad x=3u+1\quad u=1-t^4\quad t=\frac{1}{z}\)

  8. \(\displaystyle \left.\frac{dy}{dx}\right|_{x=-1}\) si \(\displaystyle y=x^6-\frac{1}{x^6}\)

  9. \(\displaystyle \frac{dy}{dx}\) si \(\displaystyle x^2y+5y^2=3x+y\)

  10. \(\displaystyle \left.\frac{dy}{dx}\right|_{x=1\;y=1}\) si \(\displaystyle 3x^3y-4y^2=5x^2y^4-6\)

Réponse

Évaluez :

  1. \(\displaystyle f'(x)=\frac{1}{x^\frac{2}{3}}-\frac{21}{16x^{\frac{7}{4}}}-x^\frac{3}{2}\)

  2. \(\displaystyle f'(1)=1+\frac{\sqrt{3}}{6}\)

  3. \(\displaystyle \frac{dy}{dx}=2x(x^2+3)^3(2x^3-5)^2(17x^3+27x-20)\)

  4. \(\displaystyle \frac{dy}{dx}=\frac{35x^4}{(7+x)^6}\)

  5. \(\displaystyle \frac{dh}{dt}=\frac{\frac{t-1}{\sqrt{t^2-2t}}-3(1-t)^2}{3\sqrt[3]{\left((1-t)^3+\sqrt{t^2-2t}\right)^2}}\)

  6. \(\displaystyle \left.\frac{dg}{dt}\right|_{t=-2}=1\)

  7. \(\displaystyle \left.\frac{dy}{dz}\right|_{z=1/2}=-\frac{1\,316\,928}{7}\)

  8. \(\displaystyle \left.\frac{dy}{dx}\right|_{x=-1}=-12\)

  9. \(\displaystyle \frac{dy}{dx}=\frac{3-2xy}{x^2+10y-1}\)

  10. \(\displaystyle \left.\frac{dy}{dx}\right|_{x=1\;y=1}=-0,04\)