Differentiate y=x*ln(x^3-5)

We can immediately see that more than differentiation rule will be needed here. The expression in question is the product of two smaller expressions, so the product rule may be useful. But to apply the product rule, we need to be able to differentiate the two smaller expressions. ln(x3-5) is slightly more complicated to differentiate. However, notice it is the composition of two functions we know how to differentiate: x3-5 and ln(x). This suggest we may be able to apply the chain rule.

First, let u=ln(x3-5)

and v=x3-5

Then u=ln(v)

Differentiating u and v:



Recall the formula for the chain rule, which in this case is du/dx=(du/dv)*(dv/dx)

Substituting into the chain rule:





So, d/dx(ln(x3-5))=3x2/(x3-5)

Note – In an exam, it may be faster simply to use the standard formula for differentiating ln: d/dx(ln(f(x)))=f'(x)/f(x) . You can use this formula whenever you spot you are differentiating ln of some function. You should be able to see how this would work in the above example. I have provided a full method for clarity, not because it is necessary to do so in your exam.

We are now in a position to apply the product rule. Recall that the formula for the product rule is d/dx(UV)=V*dU/dx+U*dV/dx   (U and V here used just to avoid confusion with u and v used earlier)

Let U=x

and V=ln(x3-5)

then dU/dx=1

and dV/dx=3x2/(x3-5)

Substituting into the product rule formula:





This gives us our answer:


Nat N. GCSE Maths tutor, A Level Maths tutor, GCSE Further Mathematic...

1 year ago

Answered by Nat, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist


£20 /hr

Mihir P.

Degree: Economics (Bachelors) - Cambridge University

Subjects offered:Maths, Economics+ 1 more

-Personal Statements-

“Hey! My name is Mihir and I'm a Cambridge Economics student. I'm a friendly, hard-working tutor that will help students gain confidence in exams.”

MyTutor guarantee

£26 /hr

Scott R.

Degree: PGCE Secondary Mathematics (Other) - Leeds University

Subjects offered:Maths, Further Mathematics

Further Mathematics

“I am currently completing 2 PGCEs in Leeds. I have always had a passion for maths and my objective is to help as many as possible reach their full potential.”

£20 /hr

Jamie M.

Degree: Physics with Theoretical Physics (Masters) - Nottingham University

Subjects offered:Maths, Physics+ 3 more

Further Mathematics
-Personal Statements-

“In my own A-Levels I developed skills that allowed me to achieve 4A*s and I want to share these techniques with any students willing to learn.”

About the author

Nat N.

Currently unavailable: until 22/06/2016

Degree: Maths (Bachelors) - Warwick University

Subjects offered:Maths, Physics+ 1 more

Further Mathematics

“Top tutor from the renowned Russell university group, ready to help you improve your grades.”

MyTutor guarantee

You may also like...

Other A Level Maths questions

How do I find the nature of a stationary point

x^2 + y^2 + 10x + 2y - 4xy = 10. Find dy/dx in terms of x and y, fully simplifying your answer.

Express 6cos(2x) + sin(x) in terms of sin(x), hence solve the equation 6cos(2x) + sin(x) = 0 for 0<x<360

How to integrate ln(x)

View A Level Maths tutors

We use cookies to improve your site experience. By continuing to use this website, we'll assume that you're OK with this. Dismiss