How do you go about differentiating a^x functions?

A key point to remember here is that a^x could mean the base (a) is not “e”-that special number which has a gradient function, dy/dx (the differential), EQUAL to the function itself, y=e^x. When "a" is any real number, you must treat it differently to functions with the base "e".

For the function y=a^x, the rule is simply dy/dx=a^x*lna

(^ means “to the power of”. * means “multiply”)

The proof for this requires an understanding of implicit differentiation (differentiating both x and y terms within an implicit relation).

Start with the function


ln (natural log) both sides of the equation



Differentiate implicitly


(the implicit part means you differentiate with respect to y but multiply by dy/dx after...)

1/y*dy/dx=lna (<----"xlna" differentiates to simply lna because a is just a constant and so lna is also just a constant)

Multiply across by y


Since y=a^x...


N.B. differentiating a^(f(x)) where f(x) is a function of x requires use of the chain rule too (first set the function f(x) to another letter, say u, so you have a^u. Differentiate a^u with the above rule. Differentiate u=f(x) separately, then use chain rule).

Adam W. GCSE Biology tutor, GCSE Chemistry tutor, GCSE Maths tutor, A...

2 years ago

Answered by Adam, an A Level Maths tutor with MyTutor

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


£20 /hr

Vlad B.

Degree: Mathematics (Masters) - Bath University

Subjects offered:Maths, Further Mathematics

Further Mathematics

“Hi! I am studying Mathematics at Bath University and I am passionate about teaching, and learning mathematics.”

MyTutor guarantee

£20 /hr

Mustafa A.

Degree: Economics (Bachelors) - Birmingham University

Subjects offered:Maths, Economics


“Highly proficient Maths and Economics tutor who's goal is to ensure that the student comes from a session with the ability to tackle the problem with ease.”

MyTutor guarantee

£22 /hr

Max G.

Degree: Mathematics and Physics (Masters) - Durham University

Subjects offered:Maths, Physics+ 1 more

Further Mathematics

“I am a maths and physics student at the University of Durham, for as long as I can remember i have been obsessed with all things science! I am patient, friendly and most of all understanding to the fact that the sciences aren't for ev...”

About the author

Adam W.

Currently unavailable:

Degree: Molecular & Cellular Biochemistry (Masters) - Oxford, Oriel College University

Subjects offered:Maths, Science+ 4 more

Extended Project Qualification
-Personal Statements-

“Hello! A warm welcome to all… My name is Adam and I am a biochemistry student at Oxford University. In brief, Biochemistry encompasses all things science but particularly requires a deep and thorough understanding of maths, organic ch...”

You may also like...

Posts by Adam

How do you go about differentiating a^x functions?

In terms of reaction mechanisms, what exactly is the rate-determining step?

To what extent do the structures and functions of DNA and RNA differ?

What is the structure and role of a chloroplast?

Other A Level Maths questions

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

What is the second derivative used for?

Find the turning point of the function y=f(x)=x^2+4x+4 and state wether it is a minimum or maximum value.

When using the addition rule in probability, why must we subtract the "intersection" to find the "union" with the Addition Rule?

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