The power, constant multiple and sum rules

Warm-up

Three quick questions from earlier lessons.

Discover

From the limit definition: $(x^2)' = 2x$ and $(x^3)' = 3x^2$ . Guess the pattern: what is $(x^4)'$ ?

$4x^3$
$x^3$

Key idea

Three rules replace the limit for every polynomial and power.

Power rule:

(x^n)' = n\,x^{n-1}

for every real

n

. Constant multiple:

(k\,f)' = k\,f'

. Sum:

(f + g)' = f' + g'

. Constant:

(c)' = 0

Worked example

$f(x) = 5x^3 - \dfrac{2}{x} + 7$ . Find $f'(x)$ .

f(x) = 5x^3 - 2x^{-1} + 7

Write the fraction as a power.

f'(x) = 15x^2 + 2x^{-2}

Power rule term by term; the constant 7 gives 0.

f'(x) = 15x^2 + \dfrac{2}{x^2}

Back to a fraction if you like.

Your turn, guided

Your turn: $f(x) = 4x^5 + 3x - 9$ .

f'(x) = 20x^4 + 3

20x⁴ from 4x⁵, 3 from 3x, 0 from −9.

Practice ★

$f(x) = 8x^{ 5 } + 3x$ . Find $f'(x)$ .
$f(x) = 4x^{ 3 } + 6x$ . Find $f'(x)$ .
$f(x) = 6x^{ 5 } + 5x$ . Find $f'(x)$ .

Watch out!

Practice ★★

$f(x) = \dfrac{ 6 }{x^{ 2 } }$ . Find $f'(x)$ .
$f(x) = \dfrac{ 9 }{x^{ 3 } }$ . Find $f'(x)$ .

$f(x) = 3\sqrt{x} + 8$ . Find $f'(x)$ .
$f(x) = 5\sqrt{x} + 4$ . Find $f'(x)$ .

Exit ticket

Three questions to check you have got it.

$f(x) = 8x^{ 5 } + 3x$ . Find $f'(x)$ .
$f(x) = 4x^{ 3 } + 6x$ . Find $f'(x)$ .
$f(x) = 6x^{ 5 } + 5x$ . Find $f'(x)$ .

Summary card

Basic rules

$(x^n)' = n x^{n-1}$ ; rewrite roots and fractions as powers first.
Constants factor out; sums split; a constant alone gives 0.

Watch out: with negative exponents, $-2 - 1 = -3$ .

End of the lesson

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