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

Question

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

Cameron Alexander · Accepted Answer

Please see below.

Consider the given function #f(x)=((x^2-5x^-1)/((5x)(-.5x)))^3(7x^6+4x^6-5x^2)# We can split it into two functions #g(x)=((x^2-5x^-1)/((5x)(-.5x)))^3# and #h(x)=7x^6+4x^6-5x^2=11x^6-5x^2#. Note that using product rule #(df)/(dx)=g'(x)h(x)+g(x)h'(x)#n and we know that #h'(x)=(dh)/(dx)=66x^5-10x#, we need to find #g'(x)#. For #g(x)# we can use chain rule, as #g(x)=(u(x))^3# and hence #(dg)/(dx)=3(u(x))^2=3((x^2-5x^-1)/((5x)(-.5x)))^2xx(du)/(dx)# and as #u(x)=(x^2-5x^-1)/((5x)(-.5x))=-(x^2-1/x)/(2.5x^2)# Now for finding #(du)/(dx)#, we can either have #u=-1/2.5+1/(2.5x^3)=-0.4+0.4x^(-3)#, which gives us #(du)/(dx)=-1.2/x^4# or use quotient formula, as follows #(du)/(dx)=-(2.5x^2(2x+1/x^2)-5x(x^2-1/x))/(6.25x^4)# = #-(5x^3+2.5-5x^3+5)/(6.25x^4)# = #-7.5/(6.25x^4)=-1.2/x^4# Hence #(dg)/(dx)=3((x^2-5x^-1)/((5x)(-.5x)))^2xx-1.2/x^4# and #(df)/(dx)=3((x^2-5x^-1)/((5x)(-.5x)))^2xx-1.2/x^4(11x^6-5x^2)+(66x^5-10x)((x^2-5x^-1)/((5x)(-.5x)))^3# and then simplified as indicated above.

Emilia Aguilar · Answer

undefined 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.