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Section Solutions

  1. Graph these functions and their derivatives.

    1. \(F'(x)=-2x\)

    2. \(h'(x) = 3x^2-9\)

    3. \(f'(x)=-2\sin(2x)\)

    4. \(g'(t)=3(2+3t)/|2+3t|\)

  2. Compute the derivative of each function.

    1. \(\dsp g'(t)=\frac{\sqrt{3}}{2\sqrt{t}}-\frac{3}{t^2} + \frac{15}{t^4}\)

    2. \(z'(t) = 5x^4+9x^2+2\)

    3. \(\dsp y' = -\frac{2x^4-13x^2-2}{(x^2-2)^2}\)

    4. \(g'(t)=48t^{11}(5t^7-2t-4)^{11}(10t^7-t-1)\)

    5. \(\dsp F'(z)=\frac{2+21z^2}{3\sqrt[3]{(2z+7z^3)^2}}\)

  3. Compute and simplify the derivatives of these exponential and logarithmic functions.

    1. \(a'(x) = -2xe^{x^2}\)

    2. \(b'(x) = 2/x\)

    3. \(c'(x) = \frac{x e^{\sqrt{x^2-1}}}{\sqrt{x^2-1}}\)

    4. \(d'(x) = 3 +\ln(x^3)\)

    5. \(\dsp g'(t)=\frac{(3t^2-16t+15)e^t - 3t^2+5}{(3t^2-10t+5)^2}\)

    6. \(\dsp f'(t) = \frac{-t}{e^{t}}\)

    7. \(n'(x) = 2x + 1/x\)

  4. Compute and simplify the derivatives of these trigonometric functions.

    1. \(C'(\beta )=3\left( \cos(5\beta) -5\beta \sin(5\beta)\right)\)

    2. \(H'(x)=(10x+1)(4x^4+\tan(x))+(5x^2+x)(16x^3+\sec^2(x))\)

    3. \(\dsp o'(x) = -\frac{\cos^2(x)+1}{\sin^3(x)}\)

    4. \(p'(x) = -\sin(x)\cos(\cos(x))\)

    5. \(\dsp r'(x) = 3\sin^2(x)\frac{x \cos(x)-\sin(x)}{x^4}\)

    6. \(i'(x) = \;\;\; \mbox{ same answer as last one!}\)

  5. Compute and simplify the derivatives of these mixed functions.

    1. \(g'(t) = te^t\cos(t) + (t+1)e^t\sin(t)\)

    2. \(h'(x) = (3x^2-3)\sec^2(x^3-3x)\)

    3. \(\dsp k'(t) = -\frac{2x+3}{\sqrt{1-(x^2+3x)^2}}\)

    4. \(\dsp m'(x) = \frac{5}{\sqrt{1-25x^2}}\)

    5. \(q'(t) = (2t-2)e^{\sec(t^2-2t)}\sec(t^2-2t)\tan(t^2-2t)\)

    6. \(s'(x) = 3x^2 + 3^x\ln(3)\)

    7. \(t'(y) = \ln(2)(2y-3)2^{y^2-3y}\)

    8. \(u'(x) = 3^{\sin(x)} + \ln(3)x\cos(x)3^{\sin(x)}\)

    9. \(G'(x)=(x+1)^{\sin(x)}\left( \cos(x)\ln(x+1) + \sin(x)/(x+1) \right)\)

  6. No solution.

  7. No solution.

  8. No solution.

  9. \(f'(0) = 2.\)

  10. \(y=(\sqrt{3} + 4\pi/3) x - 4\pi^2/9 \)

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