MYTUTOR SUBJECT ANSWERS

314 views

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:

du/dv=1/v

dv/dx=3x2

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

Substituting into the chain rule:

du/dx=(du/dv)*(dv/dx)

=(1/v)*(3x2)

=3x2/v

=3x2/(x3-5)

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:

d/dx(x*ln(x3-5))=dx(UV)

=V*dU/dx+U*dV/dx

=ln(x3-5)*1+x*3x2/(x3-5)

=ln(x3-5)+3x3/(x3-5)

This gives us our answer:

dy/dx=ln(x3-5)+3x3/(x3-5)

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

10 months ago

Answered by Nat, an A Level Maths tutor with MyTutor


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

181 SUBJECT SPECIALISTS

£20 /hr

Chris S.

Degree: Psychology (Bachelors) - Nottingham University

Subjects offered: Maths, Psychology+ 1 more

Maths
Psychology
-Personal Statements-

“Who am I? A rather broad question I know but I will do my best. I am currently a 2nd year Psychology student at the University of Nottingham and my previous education all took place in my hometown of Bristol. I have always had a passio...”

MyTutor guarantee

£20 /hr

Kirill M.

Degree: Physics (Masters) - Oxford, St Catherine's College University

Subjects offered: Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
.PAT.

“Me Hi, I'm Kirill, I'm a 3rd year physicist at Oxford. Whether you are looking for exam preparation or for a better understanding of the material that you are covering in your studies of Maths or Physics, I can help you reach your goa...”

£22 /hr

Shivali J.

Degree: Medicine (Bachelors) - Imperial College London University

Subjects offered: Maths, Chemistry+ 5 more

Maths
Chemistry
Biology
.UKCAT.
.BMAT (BioMedical Admissions)
-Personal Statements-
-Medical School Preparation-

“Who am I? I am a first year medical student at Imperial College London. I have always been curious about the world we live in and have satisified this innate drive to understand the world around us through science. I hope to share thi...”

About the author

Nat N.

Currently unavailable: until 22/06/2016

Degree: Maths (Bachelors) - Warwick University

Subjects offered: Maths, Physics+ 1 more

Maths
Physics
Further Mathematics

“Me:Hello! I am a Maths student at Warwick university”

MyTutor guarantee

You may also like...

Other A Level Maths questions

Showing all your working, evaluate ∫(21x^6 - e^2x- (1/x) +6)dx

How can we remember the difference between differentiation and integration?

If cos(x)= 1/3 and x is acute, then find tan(x).

How do I find the maximum/minimum of a function?

View A Level Maths tutors

Cookies:

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

mtw:mercury1:status:ok