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The following are examples of quotients:

Learn how to differentiate functions divided using the quotient rule in calculus, compared to products. Simplify examples step by step for better understanding.

catherinebaker
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The following are examples of quotients:

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  1. (a) (b) (c) (d) The following are examples of quotients: (c) can be divided out to form a simple function as there is a single polynomial term in the denominator. For the others we use the quotient rule.

  2. The quotient rule gives us a way of differentiating functions which are divided. The rule is similar to the product rule. where u and v are functions of x. This rule can be derived from the product rule but it is complicated. If you want to go straight to the examples, click on the box below. Examples

  3. The problem now is that this v is not the same as the v of the product rule. That v is replaced by . So, becomes Simplifying Part of the 2nd term, , is the derivative of but with respect to x not v. We can develop the quotient rule by using the product rule!

  4. We use the chain rule: So, Then, Make the denominators the same by multiplying the numerator and denominator of the 1st term by v. Write with a common denominator:

  5. e.g. 1 Differentiate to find . Solution: and We now need to simplify.

  6. We could simplify the numerator by taking out the common factor x, but it’s easier to multiply out the brackets. We don’t touch the denominator. Multiplying out numerator: Now collect like terms: and factorise: We leave the brackets in the denominator as the factorised form is simpler.

  7. e.g. can be written as In the quotient above, and In the product , and Quotients can always be turned into products. However, differentiation is usually more awkward if we do this. ( both simple functions ) ( v needs the chain rule )

  8. where u and v are both functions of x If , SUMMARY To differentiate a quotient: • Check if it is possible to divide out. If so, do it and differentiate each term. • Otherwise use the quotient rule:

  9. Exercise Use the quotient rule, where appropriate, to differentiate the following. Try to simplify your answers: 1. 2. 3. 4.

  10. and 1. Solution:

  11. and 2. Solution:

  12. and 3. Solution:

  13. 4. Solution: Divide out:

  14. = y tan x We can now differentiate the trig function by writing

  15. So, Also, is defined as So, This answer can be simplified:

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