Notes & theory for this topic

Differentiation

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1. Differentiate the following expressions.
  1. \(y = x^{4} - 8x^{3} + 4x^{2} + 3x + 1\)
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    Differentiate each term with the power rule: multiply by the power, then reduce the power by one.
    \(\frac{dy}{dx} = 4x^{3} - 24x^{2} + 8x + 3\)
  2. \(y = \sin x\)
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    The derivative of sin is cos.
    \(\frac{dy}{dx} = \cos x\)
  3. \(y = -\cos x\)
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    The derivative of cos is −sin.
    \(\frac{dy}{dx} = \sin x\)
  4. \(y = -4e^{x}\)
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    e raised to x is its own derivative.
    \(\frac{dy}{dx} = -4e^{x}\)
2. Differentiate using the chain rule.
  1. \(s = -7\sin\left(5t^{2} + 8t - 4\right)\)
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    Let \(u = 5t^{2} + 8t - 4\). Then \(\frac{ds}{du} = -7\cos u\) and \(\frac{du}{dt} = 10t + 8\); multiply them.
    \(\frac{ds}{dt} = \left(-70t - 56\right)\cos\left(5t^{2} + 8t - 4\right)\)
  2. \(s = \cos\left(6t^{2} - t + 4\right)\)
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    Let \(u = 6t^{2} - t + 4\). Then \(\frac{ds}{du} = -\sin u\) and \(\frac{du}{dt} = 12t - 1\); multiply them.
    \(\frac{ds}{dt} = \left(-12t + 1\right)\sin\left(6t^{2} - t + 4\right)\)
  3. \(s = 8e^{t^{2} + 8t - 8}\)
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    Let \(u = t^{2} + 8t - 8\). Then \(\frac{ds}{du} = 8e^{u}\) and \(\frac{du}{dt} = 2t + 8\); multiply them.
    \(\frac{ds}{dt} = \left(16t + 64\right)e^{t^{2} + 8t - 8}\)
  4. \(s = -2\ln\left(8t^{2} + 6t + 5\right)\)
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    Let \(u = 8t^{2} + 6t + 5\). Then \(\frac{ds}{du} = \frac{-2}{u}\) and \(\frac{du}{dt} = 16t + 6\); multiply them.
    \(\frac{ds}{dt} = \frac{-32t - 12}{8t^{2} + 6t + 5}\)
3. Differentiate the following products (use the product rule).
  1. \(y = \left(7x^{2} - 4x + 4\right)\left(5x^{2} - 8x - 4\right)\)
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    Product rule: \(u'v + uv'\), with \(u' = 14x - 4\) and \(v' = 10x - 8\), then collect like terms.
    \(\frac{dy}{dx} = 140x^{3} - 228x^{2} + 48x - 16\)
  2. \(y = \left(7x^{2} + 5x - 3\right)\sin\left(8x - 2\right)\)
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    Product rule: differentiate one factor at a time, keeping the other, and add.
    \(\frac{dy}{dx} = \left(14x + 5\right)\sin\left(8x - 2\right) + 8\left(7x^{2} + 5x - 3\right)\cos\left(8x - 2\right)\)
  3. \(y = \left(8x^{2} + 2x + 6\right)\cos\left(5x + 2\right)\)
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    Product rule: differentiate one factor at a time, keeping the other, and add.
    \(\frac{dy}{dx} = \left(16x + 2\right)\cos\left(5x + 2\right) - 5\left(8x^{2} + 2x + 6\right)\sin\left(5x + 2\right)\)
  4. \(y = \left(6x^{2} + 5x - 2\right)e^{3x + 7}\)
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    Product rule: differentiate one factor at a time, keeping the other, and add.
    \(\frac{dy}{dx} = \left(18x^{2} + 27x - 1\right)e^{3x + 7}\)
4. Differentiate the following quotients (use the quotient rule).
  1. \(s = \frac{t - 5}{8t - 2}\)
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    Quotient rule: \(\frac{u'v - v'u}{v^2}\), with \(u' = 1\) and \(v' = 8\).
    \(\frac{ds}{dt} = \frac{38}{\left(8t - 2\right)^{2}}\)
  2. \(s = \frac{5t^{2} - 2t + 2}{5t + 5}\)
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    Quotient rule: \(\frac{u'v - v'u}{v^2}\), with \(u' = 10t - 2\) and \(v' = 5\).
    \(\frac{ds}{dt} = \frac{25t^{2} + 50t - 20}{\left(5t + 5\right)^{2}}\)
  3. \(s = \frac{7t^{2} - 6t + 6}{8t^{2} + 2t - 1}\)
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    Quotient rule: \(\frac{u'v - v'u}{v^2}\), with \(u' = 14t - 6\) and \(v' = 16t + 2\).
    \(\frac{ds}{dt} = \frac{62t^{2} - 110t - 6}{\left(8t^{2} + 2t - 1\right)^{2}}\)
  4. \(s = \frac{\sin\left(7t - 4\right)}{\cos\left(7t - 4\right)}\)
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    Recognise this as \(\tan\left(7t - 4\right)\); its derivative is \(\sec^{2}\) of the inner expression, times the derivative of the inner expression.
    \(\frac{ds}{dt} = 7\sec^{2}\left(7t - 4\right)\)
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