Notes & theory for this topic

Differentiation

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1. Differentiate the following expressions.
  1. \(s = 5t^{4} - 2t^{3} + t^{2} + 4t + 8\)
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    Differentiate each term with the power rule: multiply by the power, then reduce the power by one.
    \(\frac{ds}{dt} = 20t^{3} - 6t^{2} + 2t + 4\)
  2. \(s = 8\sin t\)
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    The derivative of sin is cos.
    \(\frac{ds}{dt} = 8\cos t\)
  3. \(s = -6\cos t\)
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    The derivative of cos is −sin.
    \(\frac{ds}{dt} = 6\sin t\)
  4. \(s = -3e^{t}\)
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    e raised to x is its own derivative.
    \(\frac{ds}{dt} = -3e^{t}\)
2. Differentiate using the chain rule.
  1. \(y = 3\sin\left(8x^{2} + 2x - 1\right)\)
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    Let \(u = 8x^{2} + 2x - 1\). Then \(\frac{dy}{du} = 3\cos u\) and \(\frac{du}{dx} = 16x + 2\); multiply them.
    \(\frac{dy}{dx} = \left(48x + 6\right)\cos\left(8x^{2} + 2x - 1\right)\)
  2. \(y = 5\cos\left(x^{2} + 3x - 2\right)\)
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    Let \(u = x^{2} + 3x - 2\). Then \(\frac{dy}{du} = -5\sin u\) and \(\frac{du}{dx} = 2x + 3\); multiply them.
    \(\frac{dy}{dx} = \left(-10x - 15\right)\sin\left(x^{2} + 3x - 2\right)\)
  3. \(y = 4e^{6x^{2} + 5x + 7}\)
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    Let \(u = 6x^{2} + 5x + 7\). Then \(\frac{dy}{du} = 4e^{u}\) and \(\frac{du}{dx} = 12x + 5\); multiply them.
    \(\frac{dy}{dx} = \left(48x + 20\right)e^{6x^{2} + 5x + 7}\)
  4. \(y = -4\ln\left(8x^{2} - 4x + 4\right)\)
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    Let \(u = 8x^{2} - 4x + 4\). Then \(\frac{dy}{du} = \frac{-4}{u}\) and \(\frac{du}{dx} = 16x - 4\); multiply them.
    \(\frac{dy}{dx} = \frac{-64x + 16}{8x^{2} - 4x + 4}\)
3. Differentiate the following products (use the product rule).
  1. \(y = \left(5x^{2} - 4x - 2\right)\left(7x^{2} + 2x - 4\right)\)
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    Product rule: \(u'v + uv'\), with \(u' = 10x - 4\) and \(v' = 14x + 2\), then collect like terms.
    \(\frac{dy}{dx} = 140x^{3} - 54x^{2} - 84x + 12\)
  2. \(y = \left(3x^{2} - 5x + 2\right)\sin\left(8x - 5\right)\)
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    Product rule: differentiate one factor at a time, keeping the other, and add.
    \(\frac{dy}{dx} = \left(6x - 5\right)\sin\left(8x - 5\right) + 8\left(3x^{2} - 5x + 2\right)\cos\left(8x - 5\right)\)
  3. \(y = \left(x^{2} - 2x + 3\right)\cos\left(x + 5\right)\)
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    Product rule: differentiate one factor at a time, keeping the other, and add.
    \(\frac{dy}{dx} = \left(2x - 2\right)\cos\left(x + 5\right) - \left(x^{2} - 2x + 3\right)\sin\left(x + 5\right)\)
  4. \(y = \left(6x^{2} - 3x - 4\right)e^{x + 2}\)
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    Product rule: differentiate one factor at a time, keeping the other, and add.
    \(\frac{dy}{dx} = \left(6x^{2} + 9x - 7\right)e^{x + 2}\)
4. Differentiate the following quotients (use the quotient rule).
  1. \(x = \frac{8t - 3}{8t - 8}\)
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    Quotient rule: \(\frac{u'v - v'u}{v^2}\), with \(u' = 8\) and \(v' = 8\).
    \(\frac{dx}{dt} = \frac{-40}{\left(8t - 8\right)^{2}}\)
  2. \(x = \frac{2t^{2} + 3t + 8}{3t + 6}\)
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    Quotient rule: \(\frac{u'v - v'u}{v^2}\), with \(u' = 4t + 3\) and \(v' = 3\).
    \(\frac{dx}{dt} = \frac{6t^{2} + 24t - 6}{\left(3t + 6\right)^{2}}\)
  3. \(x = \frac{2t^{2} - 5t - 7}{4t^{2} - t + 4}\)
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    Quotient rule: \(\frac{u'v - v'u}{v^2}\), with \(u' = 4t - 5\) and \(v' = 8t - 1\).
    \(\frac{dx}{dt} = \frac{18t^{2} + 72t - 27}{\left(4t^{2} - t + 4\right)^{2}}\)
  4. \(x = \frac{\sin\left(3t + 2\right)}{\cos\left(3t + 2\right)}\)
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    Recognise this as \(\tan\left(3t + 2\right)\); its derivative is \(\sec^{2}\) of the inner expression, times the derivative of the inner expression.
    \(\frac{dx}{dt} = 3\sec^{2}\left(3t + 2\right)\)
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