Zestaw 6

I.Określić pochodną I obliczyć znaczenie w punkcie.

1. y= ln(sin(4x)) ,x=$\frac{x}{16}$

$\frac{\text{dy}}{\text{dx}}$= 4 ctg (4x) $\frac{\text{dy}}{\text{dx}}$ _x=$\frac{\pi}{16}$ =4

2. y=cos²(x²) ,x=$\frac{\sqrt{\pi}}{2}$

$\frac{\text{dy}}{\text{dx}}$= -2 xsin(2x²) $\frac{\text{dy}}{\text{dx}}$ _x=$\frac{\sqrt{\pi}}{2}$ =-$\sqrt{\pi}$ ≈-1.772

3. y=e^sin(2x)-3 cos(2x) ,x=0

$\frac{\text{dy}}{\text{dx}}$=2e^sin(2x)cos(2x)+6 sin(2x) $\frac{\text{dy}}{\text{dx}}$ _x=0 =2

4. y=tg²(3x) ,x=0

$\frac{\text{dy}}{\text{dx}}$ =$\frac{6\ tg(3x)}{\cos^{2}(3x)}$ $\frac{\text{dy}}{\text{dx}}$ _x=0 =0

5. y=$\frac{\sqrt{x^{2} + 1}}{x}$ ,x=$\sqrt{3}$

$\frac{\text{dy}}{\text{dx}}$ =- $\frac{1}{x^{2}\sqrt{x^{2} + 1}}$ $\frac{\text{dy}}{\text{dx}}$ _x=$\sqrt{3}$ = -$\frac{1}{6}$

II.Określić pochodną funkcji zadanej parametrycznie, obliczyć znaczenie w punkcie.

6. y=$\sqrt{t}$+3sin(t) ,x=ln(cos(t)),t=1

$\frac{\text{dy}}{\text{dt}}$ =$\frac{1}{2\sqrt{t}}$ + 3cos(t) $\frac{\text{dx}}{\text{dt}}$ = - tg(t)

$\frac{\text{dy}}{\text{dx}}$ = -ctg(t) ($\frac{1}{2\sqrt{t}}$ + 3cos(t))

$\frac{\text{dy}}{\text{dx}}$ _t=1 =-ctg(1)($\frac{1}{2}$ +3cos(1))≈-1.362

7. y=tg($\sqrt{t}$) ,x=e^{−2t2 + 1} ,t=1

$\frac{\text{dy}}{\text{dt}}$ =$\frac{1}{2\sqrt{\text{t\ }}\cos^{2}(\sqrt{t})}$ $\frac{\text{dx}}{\text{dt}}$ =-4te^{−2t2 + 1}

$\frac{\text{dy}}{\text{dx}}$ = -$\frac{e^{{2t}^{2} - 1}}{8\sqrt{t^{3}}\cos^{2}(\sqrt{t})}$ $\frac{\text{dy}}{\text{dx}}$ _t=1 =-$\frac{e}{8\cos^{2}(1)}$ ≈-1.164

8. y=2 sin(t) ,x=(t+2) cos(t) ,t=$\frac{\pi}{4}$

$\frac{\text{dy}}{\text{dt}}$=2 cos(t) $\frac{\text{dx}}{\text{dt}}$= cos(t)-(t+2) sin (t)

$\frac{\text{dy}}{\text{dx}}$= $\frac{2}{1 - \left( t + 2 \right)tg\ (t)}$ $\frac{\text{dx}}{\text{dt}}$ _t=$\frac{\pi}{4}$ = -$\frac{8}{\pi + 4}$ ≈-1.120

9. e^xy-2xy+y=1 ,f(x,y)=e^xy-2xy+y-1

$\frac{\partial f(x,y)}{\partial y}$=xe^xy-2x+1 $\frac{\partial f(x,y)}{\partial x}$=ye^xy-2y

$\frac{\text{dy}}{\text{dx}}$=-$\frac{\frac{\partial f(x,y)}{\partial x}}{\frac{\partial f(x,y)}{\partial y}}$ =$\frac{2y - \text{ye}^{\text{xy}}}{\text{xe}^{\text{xy}} - 2x + 1}$

10. $\frac{x^{2}}{2}$ -e^x2 + 1+y=0 ,f(x,y)=$\frac{x^{2}}{2}$ - e^x2 + 1+y

$\frac{\partial f(x,y)}{\partial y}$=1 $\frac{ef(x,y)}{\partial x}$ = x (1-2e^x2 + 1)

$\frac{\text{dy}}{\text{dx}}$ = -$\frac{\frac{\partial f(x,y)}{\partial x}}{\frac{\partial f(x,y)}{\partial y}}$ = x(2e^x2 + 1-1)

Własności rachunku różniczkowego:

1. y=c$\overset{\Rightarrow}{\ }$y’=0

2. y=cf(x) $\overset{\Rightarrow}{\ }$ y’=cf’(x)

3. y=f(x) +g(x) $\overset{\Rightarrow}{\ }$ y’ =f’(x) + g’(x)

4. y=f(x)g(x) $\overset{\Rightarrow}{\ }$ y’ =f’(x)g(x)+f(x)g’(x)

5. y=$\frac{f(x)}{g(x)}$ $\overset{\Rightarrow}{\ }$y’=$\frac{f^{'}\left( x \right)g\left( x \right) - f\left( x \right)g^{'}\left( x \right)}{g{(x)}^{2}}$

6. y=f(g(x)) $\overset{\Rightarrow}{\ }$ $\frac{\text{dy}}{\text{dx}}$ =($\frac{\text{df}(t)}{\text{dt}}$)_t=g(x) $\frac{dg(x)}{\text{dx}}$

7. y=f(t) ,x=g(t) $\overset{\Rightarrow}{\ }$ $\frac{\text{dy}}{\text{dx}}$ = $\frac{\frac{\text{dy}}{\text{dt}}}{\frac{\text{dx}}{\text{dt}}}$

8. f(x,y) =0 $\overset{\Rightarrow}{\ }\frac{\text{dy}}{\text{dx}}$ = -$\frac{\frac{\partial f(x,y)}{\partial x}}{\frac{\partial f(x,y)}{\partial y}}$

Wzory rachunku różniczkowego:

9. (x^a)’ = ax^a-1

10. (sin(x))’ = cos(x)

11. (cos(x))’ = - sin(x)

12. (tg(x))’ = $\frac{1}{\cos^{2}(x)}$

13. (e^x)’ = e^x

14. ln(x)=$\frac{1}{x}$