Differentiating with the power and sum rules

The power rule is the basis of almost every derivative. With the sum and constant multiple rules you can differentiate any polynomial, and with a little rewriting roots and fractions too.

Rules

Power rule: $(x^{n})^{'} = n x^{n - 1}$
Constant multiple: $(a \cdot f)^{'} = a \cdot f^{'}$
Sum rule: $(f + g)^{'} = f^{'} + g^{'}$
Constant: $(c)^{'} = 0$
Rewrite: $x = x^{\frac{1}{2}}$ , $\frac{1}{x ^{n}} = x^{- n}$

Your turn

Three problems on this topic. Type like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x).

Task 1 of 3Power & sum rule

Task

Differentiate:

f (x) = - 2 x^{4} + 3 x^{2} - 8 x - 9

$f^{'} (x) =$

Write like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x), sin(x). Use brackets: e^(2x), not e^2x. Numbers also as 3/4 or e^2 - 1.

Worked example

Differentiate: $f (x) = x^{3} - 6 x^{2} - 3 x$

Worked solution

1.Sum rule: differentiate each term on its own.
2.Power rule: $(a x^{n})^{'} = a \cdot n x^{n - 1}$ ; a constant drops out.
3. $f^{'} (x) = 3 x^{2} - 12 x - 3$

Typical mistake: Forgetting to lower the exponent by 1 or keeping the constant.

Differentiate: $f (x) = 4 x + \frac{3}{x ^{2}}$ (for $x > 0$ )

Worked solution

1.Write as powers: $x = x^{\frac{1}{2}}$ , $\frac{1}{x ^{2}} = x^{- 2}$ .
2.Apply the power rule to both terms.
3. $f^{'} (x) = \frac{2}{x} - \frac{6}{x ^{3}}$

Typical mistake: With $x^{- 2}$ , raising the exponent instead of lowering it: it is $- 2 - 1 = - 3$ .

Common questions

How do I differentiate a square root?

Write the root as a power with a fractional exponent, √x = x^(1/2), and use the power rule: 1/2 · x^(−1/2) = 1 / (2√x).

How do I differentiate 1/x?

1/x = x^(−1), so the derivative is −1 · x^(−2) = −1/x². The exponent drops by 1, from −1 to −2.

e^x, ln, sin, cos Antiderivatives Chain rule

Differentiating with the power and sum rules

Rules

Your turn

How do I differentiate a square root?

How do I differentiate 1/x?

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