Notes & theory for this topic

Differentiation

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1. Differentiate the following expressions.
  1. \(x = 6t^{4} - 3t^{3} + 2t^{2} - 4t - 8\)
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    Differentiate each term with the power rule: multiply by the power, then reduce the power by one.
    \(\frac{dx}{dt} = 24t^{3} - 9t^{2} + 4t - 4\)
  2. \(x = -6\sin t\)
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    The derivative of sin is cos.
    \(\frac{dx}{dt} = -6\cos t\)
  3. \(x = 8\cos t\)
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    The derivative of cos is −sin.
    \(\frac{dx}{dt} = -8\sin t\)
  4. \(x = 3e^{t}\)
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    e raised to x is its own derivative.
    \(\frac{dx}{dt} = 3e^{t}\)
2. Differentiate using the chain rule.
  1. \(x = -6\sin\left(3t^{2} - t - 2\right)\)
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    Let \(u = 3t^{2} - t - 2\). Then \(\frac{dx}{du} = -6\cos u\) and \(\frac{du}{dt} = 6t - 1\); multiply them.
    \(\frac{dx}{dt} = \left(-36t + 6\right)\cos\left(3t^{2} - t - 2\right)\)
  2. \(x = -3\cos\left(3t^{2} + 4t + 6\right)\)
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    Let \(u = 3t^{2} + 4t + 6\). Then \(\frac{dx}{du} = 3\sin u\) and \(\frac{du}{dt} = 6t + 4\); multiply them.
    \(\frac{dx}{dt} = \left(18t + 12\right)\sin\left(3t^{2} + 4t + 6\right)\)
  3. \(x = -7e^{7t^{2} + 2t + 8}\)
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    Let \(u = 7t^{2} + 2t + 8\). Then \(\frac{dx}{du} = -7e^{u}\) and \(\frac{du}{dt} = 14t + 2\); multiply them.
    \(\frac{dx}{dt} = \left(-98t - 14\right)e^{7t^{2} + 2t + 8}\)
  4. \(x = -2\ln\left(7t^{2} - t + 8\right)\)
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    Let \(u = 7t^{2} - t + 8\). Then \(\frac{dx}{du} = \frac{-2}{u}\) and \(\frac{du}{dt} = 14t - 1\); multiply them.
    \(\frac{dx}{dt} = \frac{-28t + 2}{7t^{2} - t + 8}\)
3. Differentiate the following products (use the product rule).
  1. \(y = \left(4x^{2} - 8x - 5\right)\left(x^{2} + 8x - 3\right)\)
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    Product rule: \(u'v + uv'\), with \(u' = 8x - 8\) and \(v' = 2x + 8\), then collect like terms.
    \(\frac{dy}{dx} = 16x^{3} + 72x^{2} - 162x - 16\)
  2. \(y = \left(5x^{2} + 2x - 4\right)\sin\left(8x - 8\right)\)
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    Product rule: differentiate one factor at a time, keeping the other, and add.
    \(\frac{dy}{dx} = \left(10x + 2\right)\sin\left(8x - 8\right) + 8\left(5x^{2} + 2x - 4\right)\cos\left(8x - 8\right)\)
  3. \(y = \left(3x^{2} + 6x - 7\right)\cos\left(8x + 3\right)\)
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    Product rule: differentiate one factor at a time, keeping the other, and add.
    \(\frac{dy}{dx} = \left(6x + 6\right)\cos\left(8x + 3\right) - 8\left(3x^{2} + 6x - 7\right)\sin\left(8x + 3\right)\)
  4. \(y = \left(2x^{2} - 3x + 8\right)e^{2x - 4}\)
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    Product rule: differentiate one factor at a time, keeping the other, and add.
    \(\frac{dy}{dx} = \left(4x^{2} - 2x + 13\right)e^{2x - 4}\)
4. Differentiate the following quotients (use the quotient rule).
  1. \(x = \frac{6t + 8}{t - 3}\)
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    Quotient rule: \(\frac{u'v - v'u}{v^2}\), with \(u' = 6\) and \(v' = 1\).
    \(\frac{dx}{dt} = \frac{-26}{\left(t - 3\right)^{2}}\)
  2. \(x = \frac{2t^{2} - 7t - 3}{5t - 1}\)
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    Quotient rule: \(\frac{u'v - v'u}{v^2}\), with \(u' = 4t - 7\) and \(v' = 5\).
    \(\frac{dx}{dt} = \frac{10t^{2} - 4t + 22}{\left(5t - 1\right)^{2}}\)
  3. \(x = \frac{3t^{2} - 8t - 7}{8t^{2} - 8t + 7}\)
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    Quotient rule: \(\frac{u'v - v'u}{v^2}\), with \(u' = 6t - 8\) and \(v' = 16t - 8\).
    \(\frac{dx}{dt} = \frac{40t^{2} + 154t - 112}{\left(8t^{2} - 8t + 7\right)^{2}}\)
  4. \(x = \frac{\sin\left(4t - 2\right)}{\cos\left(4t - 2\right)}\)
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    Recognise this as \(\tan\left(4t - 2\right)\); its derivative is \(\sec^{2}\) of the inner expression, times the derivative of the inner expression.
    \(\frac{dx}{dt} = 4\sec^{2}\left(4t - 2\right)\)
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