How Do You Differentiate f(x) = ax^n?

To differentiate $f(x)=ax^n$, multiply the coefficient $a$ by the exponent $n$, then subtract one from the exponent:

$$f'(x)=anx^{n-1}.$$

The Power Rule

This is the power rule. In IB Mathematics: Applications and Interpretation SL 5.3: Introduction to derivatives, $a$ is a constant and $n$ is an integer. The rule also applies term by term to polynomials.

Use these two steps:

1. Multiply the coefficient by the original exponent: $a\times n=an$.
2. Reduce only the exponent by one: $n\rightarrow n-1$.

For example, if $f(x)=4x^3$, then

$$f'(x)=4(3)x^{3-1}=12x^2.$$

The derivative (f'(x)) is the gradient function, meaning it gives the instantaneous rate of change of $f$ at each value of $x$. At $x=2$, the gradient is

$$f'(2)=12(2)^2=48.$$

Therefore, the tangent has gradient $48$ when $x=2$.

Common Cases

A linear term can be written as $6x^1$, so its derivative is $6(1)x^0=6$. A constant has derivative zero because it does not change.

The rule remains valid for zero and negative integer exponents. For example, if $g(x)=3x^{-4}$, then $g'(x)=-12x^{-5}$. When , , so the derivative is zero. Keep the sign of a negative exponent and simplify only after applying the rule.

For a polynomial, differentiate every term separately. If

$$f(x)=2x^3-5x^2+4x-7,$$

then

$$f'(x)=6x^2-10x+4.$$

This term-by-term method matters because a polynomial’s derivative combines the rate of change of each term. Always copy every term into your working first; this reduces sign errors and helps the examiner follow your method clearly.

A common misconception is to subtract one from the coefficient. This is incorrect: multiply the coefficient by the original exponent, and subtract one only from the exponent. Another error is forgetting that $x=x^1$.

Exam Technique

The command term differentiate requires a correctly simplified derivative. Although every Math AI paper permits a GDC, show the power-rule working rather than only calculator output. Differentiate constants to zero, preserve negative exponents, and use (f'(x)) or $\frac{dy}{dx}$ consistently. This SL content can be assessed in Paper 1 or Paper 2.