| So far, we know how to differentiate functions like sin x and x3- 5. But how do we differentiate a function of a function? That is how can we differentiate sin (x3- 5)? |
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| This differentiation of function of a function is known as 'chain rule'. The chain rule is probably the most widely used differentiation rule in mathematics. |
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| We state the chain rule as follows: |
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| If y=f(u) and u=g(x) are differentiable functions of u and x respectively, then |
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| In other words, |
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| if y is a differentiable function of u and u is a differentiable function of x, then the derivative of y with respect to x is equal to the derivative of y with respect to u times the derivative of u with respect to x. |
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| Example 1: |
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| Suggested answer: |
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| Let y = log u, u = tan t, t = x/2 |
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| From the above example, it is clear that chain rule can be extended. |
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| Let v = f(u), u = g(x), x = h(t) |
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| Then, we have |
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| Example 2: |
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| Suggested answer: |
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| Let Dx be the increment of x and Du be the corresponding increment in u. |
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| Consider the function y = eu. |
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| Let Dy be the corresponding increment in y for the increment Du in u. |
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| = eu(1) = eu ……..(2) |
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| Therefore, we get |
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