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CHAIN RULE

CHAIN RULE. How to differentiate composite function. The differentiation formulas you learned by now do not enable you to calculate F ’( x ). It turns out that the derivative of the composite function f ◦ g is the product of the derivatives of f and g .

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CHAIN RULE

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  1. CHAIN RULE How to differentiate composite function • The differentiation formulas you learned by now do not enable you to calculate F’(x). It turns out that the derivative of the composite function f ◦ g is the product of the derivatives of f and g. Proof goes over the head, so forget about that. This fact is one of the most important of the differentiation rules. It is called the Chain Rule.

  2. It is convenient if we interpret derivatives as rates of change. • Regard: • as the rate of change of u with respect to x • as the rate of change of y with respect to u • as the rate of change of y with respect to x If u changes twice as fast as x and y changes three times as fast as u, it seems reasonable that y changes six times as fast as x. So,we expect that:

  3. Chain Rule • If is differentiable at and is differentiable at , the composite function defined by is differentiable at andis given by the product: The Chain Rule can be written either in the prime notation or, if y = f(u) and u = g(x), in Leibniz notation: easy to remember because, if dy/du and du/dx were quotients, then we could cancel du. However, remember: du/dx should not be thought of as an actual quotient

  4. How to differentiate composite function Let’s go back: In order not to make your life too complicated (it is already enough), we’ll introduce one way that is most common and anyway, everyone ends up with that one: Leibnitz Let where • dy/dx refers to the derivative of y when y is considered as a function of x • (called the derivative of y with respect to x) • dy/du refers to the derivative of y when considered as a function of u • (the derivative of y with respect to u)

  5. example: • Differentiate: • )

  6. example: Differentiate y = (x3 – 1)100 Taking u= x3 – 1 and y = u100

  7. example: Find f’ (x) if • First, rewrite f as f(x) = (x2 + x + 1)-1/3 • Thus,

  8. example: Find the derivative of • Combining the Power Rule, Chain Rule, and Quotient Rule, we get:

  9. example: • Differentiate: y = (2x + 1)5 (x3 – x + 1)4 • In this example, we must use the Product Rule before using the Chain Rule.

  10. The reason for the name ‘Chain Rule’ becomes clear when we make a longer chain by adding another link. Suppose that y = f(u), u = g(x), and x = h(t), where f, g, and h are differentiable functions, then, to compute the derivative of y with respect to t, we use the Chain Rule twice:

  11. example: • Notice that we used the Chain Rule twice.

  12. example: Differentiate

  13. Divide both sides by The slope of a parametrized curve is given by: The chain rule enables us to find the slope of parametrically defined curves x = x(t) and y = y(t):

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