Chapter 4 Problems

Thirty chapter problems: 10 from the original book and 20 added for more practice.

Read the chapter problems → · Chapter solutions →

1. Original problem

   a) Implicitly differentiate the equation $y=x\cdot y\cdot e^x+1$ ​
   b) Write its derivative as a function of just x.

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2. Original problem

   Differentiate $f(x)=\frac{\tan x}{x^2-4}$

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3. Original problem

   Find the derivative of $y=\frac{e^x-4x}{{\sin }^{2}x}$

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4. Original problem

   Using the equation $\frac{(x-4{)}^2}{25}+\frac{(y+1{)}^2}{16}=1$, find the slope of the tangent line at x=1 in the first quadrant .

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5. Original problem

   Differentiate $y=\frac{xy+3e^x}{y^2}$

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6. Original problem

   Differentiate $y=\frac{x\sin y+2\cos (2x^2)}{x^3}$

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7. Original problem

   Differentiate $y=x^2\cdot \sin y\cdot e^{\cos x}$

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8. Original problem

   Using the equation given below, find the slope of its tangent line at $x=\frac{\pi }{4}$ (Ref: Unit Circle):$\begin{array}{rl}{y}^2+1& =y\cdot \sin x+y\cdot \cos (2x)+\frac{\sin x}{y}\\ & +\frac{{\cos }^{2}x}{y}-\frac{{\sin }^{2}x}{y}\end{array}$

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9. Original problem

   Find the implicit derivative of $\frac{xy+x^2}{e^{2x}}=\frac{\cos (4x^2+1)\sin y}{e^x}$

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10. Original problem

    Find the implicit derivative of $y=x\sin (\ln (y))+\frac{xy}{\cos x}$

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11. Added problem

    Differentiate $f\left(x\right)=(x^2+1)(x-2)$.

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12. Added problem

    Differentiate $f\left(x\right)=x^2\sin x$.

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13. Added problem

    Differentiate $f\left(x\right)=x{e}^x\sin x$.

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14. Added problem

    Differentiate $f\left(x\right)=(2x-1)/(x+2)$.

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15. Added problem

    Differentiate $f\left(x\right)=\sin x/x$.

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16. Added problem

    Differentiate $f\left(x\right)=(x^2+3{)}^{10}$.

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17. Added problem

    Differentiate $f\left(x\right)={\sin }^3\left(2x\right)$.

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18. Added problem

    Differentiate $f\left(x\right)=e^{-x^2}$.

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19. Added problem

    Find the derivative and its domain for $f\left(x\right)=\sqrt{3x+1}$.

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20. Added problem

    Find the tangent line to $x^2+y^2=25$ at $(3,4)$.

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21. Added problem

    Find dy/dx for $x^2+xy+y^2=7$.

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22. Added problem

    Differentiate $\sin \left(xy\right)=x$ implicitly.

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23. Added problem

    Find $f\prime \left(0\right)$ for $f\left(x\right)=(x+1)e^{2x}\cos x$.

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24. Added problem

    At $x=2$, a table gives $f=3$, $f\prime =-1$, $g=5$, and $g\prime =2$. Find the derivative of $h=fg$ there.

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25. Added problem

    At $x=2$, $f=3$, $f\prime =-1$, $g=5$, and $g\prime =2$. Find the derivative of $h=f/g$ there.

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26. Added problem

    If $g\left(1\right)=4$, $g\prime \left(1\right)=4$, and $f\prime \left(4\right)=-3$, find the derivative of $h\left(x\right)=f\left(g\right(x\left)\right)$ at 1.

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27. Added problem

    A student says the derivative of $\sin \left(3x^2\right)$ is $\cos \left(3x^2\right)$. Diagnose the error.

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28. Added problem

    What happens to the tangent of $x^2+y^2=1$ at $(1,0)$?

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29. Added problem

    Find the second derivative of $f\left(x\right)=e^{x^2}$.

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30. Added problem

    Find the slope of $x+y+xy=3$ at $(1,1)$.

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