7.2 Differentiation from first principles — Learn

Prerequisites: expanding brackets such as $(x + h)^2$; simplifying an algebraic fraction; the limit definition of the derivative from Section 7.1.

By the end of this section you can: use the definition $f'(x) = \lim_{h \to 0} \dfrac{f(x + h) - f(x)}{h}$ to differentiate a whole-number power of $x$; carry out the four steps of the method in order; and explain why $h$ is not set to zero until the division is complete.

1. The definition, one more time

Section 7.1 introduced the derivative as the limit of a chord gradient:

Differentiating from first principles means working out this limit with algebra, not with a table of numbers. Four steps do the job every time.

1. Write down $f(x + h)$.
2. Subtract $f(x)$.
3. Divide by $h$.
4. Let $h$ tend to zero.

The order matters. Step 3 must happen before step 4, because $h$ appears in the denominator.

Example 1. Differentiate $f(x) = x^2$ from first principles.

Write $f(x + h) = (x + h)^2 = x^2 + 2xh + h^2$.

Subtract $f(x) = x^2$:

Divide by $h$:

The division is safe because $h \neq 0$ while we take the limit. Let $h \to 0$:

So the derivative of $x^2$ is $2x$. At $x = 2$ this gives $4$, which agrees with the estimate in Section 7.1.

Example 2. Differentiate $f(x) = x^3$ from first principles.

Expand $(x + h)^3$ using the binomial expansion:

Subtract $f(x) = x^3$:

Divide by $h$:

Let $h \to 0$. Every term still containing $h$ vanishes:

2. A constant multiple and a sum

The method works just as well when the function has more than one term.

Example 3. Differentiate $f(x) = x^2 + x$ from first principles.

Subtract $f(x) = x^2 + x$:

Divide by $h$:

Let $h \to 0$:

Example 4. A constant function has a horizontal graph. Differentiate $f(x) = 7$ from first principles.

Here $f(x + h) = 7$ for every $h$, so $f(x + h) - f(x) = 7 - 7 = 0$. Dividing by $h$ gives $0$. The limit is $0$, so $f'(x) = 0$. A horizontal line has gradient zero.

The same argument gives the derivative of any constant: it is zero.

3. Differentiating sine and cosine from first principles

The same definition gives the derivatives of $\sin x$ and $\cos x$. Use the addition formula from Chapter 5:

Then

As $h$ tends to zero, $\dfrac{\sin h}{h}$ tends to $1$ and $\dfrac{\cos h - 1}{h}$ tends to $0$. Therefore

A similar argument starting from $\cos(x + h) = \cos x\cos h - \sin x\sin h$ gives

The minus sign often goes missing. Section 7.3 uses these results in calculations.

Watch out. Three mistakes are common in this method.

Setting $h = 0$ before dividing. If you put $h = 0$ into $f(x + h) - f(x)$ you get $0$, and then $\dfrac{0}{0}$ is meaningless. Divide by $h$ first, cancel it, and only then let $h$ tend to zero.

Expanding $(x + h)^n$ as $x^n + h^n$. The bracket is a sum, not a product. For example, $(x + h)^2 = x^2 + 2xh + h^2$, and the middle term is essential. Use the binomial expansion or multiply out carefully.

Forgetting a term after dividing. Every term that still contains $h$ goes to zero in the limit. In Example 2 the terms $3xh$ and $h^2$ both vanish, leaving only $3x^2$. Re-check: after taking the limit, no $h$ should remain in the answer.

You should now be able to

• state the definition of the derivative as a limit;
• write down $f(x + h)$ for a polynomial;
• carry out the four steps of differentiation from first principles;
• differentiate $x^2$, $x^3$ and simple sums and multiples this way;
• explain why the constant function has derivative zero;
• state the derivatives of $\sin x$ and $\cos x$ from first principles.