# How do you derive a function composed of a division and a multiplication? Do you use the quotient rule or the product rule?

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Ex:#
f(x) = ((x^2-5x^-1)/((5x)(-.5x)))^3 (7x^6+4x^6-5x^2)#

Ex:

Please see below.

and then simplified as indicated above.

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To derive a function composed of a division and a multiplication, you would typically use both the quotient rule and the product rule, depending on the specific function. If the function involves a quotient of two functions, you would use the quotient rule to differentiate it. The quotient rule states that if you have a function ( f(x) ) divided by another function ( g(x) ), the derivative ( (f/g)' ) is given by:

[ \left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2} ]

If the function involves the product of two functions, you would use the product rule to differentiate it. The product rule states that if you have a function ( f(x) ) multiplied by another function ( g(x) ), the derivative ( (fg)' ) is given by:

[ (fg)' = f'g + fg' ]

In some cases, you may need to use both the quotient rule and the product rule in succession if the function involves a combination of division and multiplication. It depends on the specific form of the function and how it can be manipulated algebraically to simplify the differentiation process.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- If #f(x)= sin6 x # and #g(x) = e^(3+2x ) #, how do you differentiate #f(g(x)) # using the chain rule?
- How do you differentiate #f(x)=x/(x+3)^2# using the quotient rule?
- How do you differentiate #f(x)= e^x/(e^(x^2) +2x )# using the quotient rule?
- How do you use the chain rule to differentiate #y=e^(sinx)#?
- What is the derivative of #f(x)=-2x^-3+x^2-7#?

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