Working out the derivative of x when the function is divided by x

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raveyp

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I've come across derivative questions like this before and I can't figure out how to get the answer (see attached files)? Am I missing a rule or something? I always work it out to be 20x using the chain rule (2*10x^1/1*x^0).

Many thanks,


12704

12703
 
It's okay. Figured it out: it should be 1*10x^0 + -1*40x^-2 = 10 - 40x^-2 = 10 - 40/x^2
 
raveyp said:
… using the chain rule (2*10x^1/1*x^0).
Click to expand...
Hello raveyp. Did you intend to say the Power Rule? (The given function is not a composition of functions, so the Chain Rule doesn't apply.) What you've applied above is the Power Rule, but that won't work on a ratio of polynomials.

You could use the Quotient Rule. Or, rewrite the given ratio as a product, and then use the Product Rule.

y = (1/x)(10x^2 + 40)

y' = (1/x)(20x) + (10x^2 + 40)(-1/x^2)

Simplify …

?
 
raveyp said:
It's okay. Figured it out: it should be 1*10x^0 + -1*40x^-2 = 10 - 40x^-2 = 10 - 40/x^2
Click to expand...
Presumably you did this by carrying out the division to find that the given function is y = 10x - 40x^-1, and then differentiating. That is the easiest way in this case, and is what I'd do. The quotient rule and product rule are worth learning for more complicated problems -- and especially to teach you that you can't just differentiate the numerator and denominator separately, as many students initially think they can do.
 
Otis said:
Hello raveyp. Did you intend to say the Power Rule? (The given function is not a composition of functions, so the Chain Rule doesn't apply.) What you've applied above is the Power Rule, but that won't work on a ratio of polynomials.

You could use the Quotient Rule. Or, rewrite the given ratio as a product, and then use the Product Rule.

y = (1/x)(10x^2 + 40)
Otis said:
Hello raveyp. Did you intend to say the Power Rule? (The given function is not a composition of functions, so the Chain Rule doesn't apply.) What you've applied above is the Power Rule, but that won't work on a ratio of polynomials.

You could use the Quotient Rule. Or, rewrite the given ratio as a product, and then use the Product Rule.

y = (1/x)(10x^2 + 40)

y' = (1/x)(20x) + (10x^2 + 40)(-1/x^2)

Simplify …

?
Click to expand...


y' = (1/x)(20x) + (10x^2 + 40)(-1/x^2)

Simplify …

?
Click to expand...


Yeah sorry *Power Rule

Ahh yes of course - multiply the whole thing by 1/x (i.e. multiply by the reciprocal, duh!)! Thanks :)
 
Dr.Peterson said:
Presumably you did this by carrying out the division to find that the given function is y = 10x - 40x^-1, and then differentiating. That is the easiest way in this case, and is what I'd do. The quotient rule and product rule are worth learning for more complicated problems -- and especially to teach you that you can't just differentiate the numerator and denominator separately, as many students initially think they can do.
Click to expand...

Yes, I believe this is what I did and I think trying to differentiate the numerator and denominator separately was where I was going wrong in the first instance. Hopefully, I'll come onto the quotient rule and product rule later on (just getting started in calculus rn).
 
And I think the question should be framed: Working out the derivative of 'f(x)' when the function is divided by x?
 
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