Assignments

The central theme of the following assignments is to calculate the derivative of an implicitly defined function $y(x)$ .

1. Calculate $y'$ if:

$(2x+1)(1+y)-1=0$

2. Calculate $y'$ if:

$(x^3-1)(1+y)=3$

3. Calculate $y'$ if:

$\displaystyle\frac{x}{y+1}=2$

4. Calculate $y'$ if:

$x\ln(y)-4=0$

5. Calculate the tangent line at the point $(0,0)$ of the graph of the following expression:

$(x+1)e^y-1=0$

6. Calculate the tangent line at the point $(1,3)$ of the graph of the following expression:

$2x+3y^2=29$

7. Calculate $y''$ if:

$xy=5$

8. If $y$ is a differentiable function of $x$ satisfying the following equation:

$2x^2+6xy+y^2=18$

calculate $y'$ .

9. Calculate the tangent line at the point $(1,4)$ of the graph of the following expression:

$-x+3^{\sqrt{y}}=8$

10. The equation:

$x^2y^3+(y+1)e^{-x}=x+2$

defines $y$ as a differentiable function of $x$ close to the point $(x,y)=(0,1)$ .

Calculate the tangent line of the graph at this point.

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