Power rule for differentiation

The power rule allows us to easy differentiate any function, as long as we only have powers of x (i.e. no addition, subtraction, multiplication or division of x terms).

To differentiate a term ax^n:

Multiply the term by the power n: a \times n.
Decrease the power by 1: n - 1.
So the derivative of ax^n is: a \times n \times x^{n-1}.

Differentiating larger expressions

Example: Differentiate f(x) = 3x^4 + 2x^3 - 5x^2 + 7x - 4

Differentiate each term separately:
- 3x^4: Multiply by 4 and decrease power by 1: 3 \times 4 \times x^{4-1} = 12x^3.
- 2x^3: Multiply by 3 and decrease power by 1: 2 \times 3 \times x^{3-1} = 6x^2.
- -5x^2: Multiply by 2 and decrease power by 1: -5 \times 2 \times x^{2-1} = -10x.
- 7x: Multiply by 1 and decrease power by 1: 7 \times 1 \times x^{1-1} = 7.
- -4: Constant term, derivative is 0.
Add together the results:
- f'(x) = 12x^3 + 6x^2 - 10x + 7.
Answer: f'(x) = 12x^3 + 6x^2 - 10x + 7.

Example: Differentiate g(x) = 5x^5 - 3x^4 + x^2 - 8

Differentiate each term separately:
- 5x^5: Multiply by 5 and decrease power by 1: 5 \times 5 \times x^{5-1} = 25x^4.
- -3x^4: Multiply by 4 and decrease power by 1: -3 \times 4 \times x^{4-1} = -12x^3.
- x^2: Multiply by 2 and decrease power by 1: 1 \times 2 \times x^{2-1} = 2x.
- -8: Constant term, derivative is 0.
Add together the results:
- g'(x) = 25x^4 - 12x^3 + 2x.
Answer: g'(x) = 25x^4 - 12x^3 + 2x.

Example: Differentiate h(x) = 4x^3 + 6x - 9

Differentiate each term separately:
- 4x^3: Multiply by 3 and decrease power by 1: 4 \times 3 \times x^{3-1} = 12x^2.
- 6x: Multiply by 1 and decrease power by 1: 6 \times 1 \times x^{1-1} = 6.
- -9: Constant term, derivative is 0.
Add together the results:
- h'(x) = 12x^2 + 6.
Answer: h'(x) = 12x^2 + 6.

Example: Differentiate k(x) = 7x^6 - 2x^3 + 5x^2 - x + 1

Differentiate each term separately:
- 7x^6: Multiply by 6 and decrease power by 1: 7 \times 6 \times x^{6-1} = 42x^5.
- -2x^3: Multiply by 3 and decrease power by 1: -2 \times 3 \times x^{3-1} = -6x^2.
- 5x^2: Multiply by 2 and decrease power by 1: 5 \times 2 \times x^{2-1} = 10x.
- -x: Multiply by 1 and decrease power by 1: -1 \times 1 \times x^{1-1} = -1.
- 1: Constant term, derivative is 0.
Add together the results:
- k'(x) = 42x^5 - 6x^2 + 10x - 1.
Answer: k'(x) = 42x^5 - 6x^2 + 10x - 1.

Differentiating terms with negative or fractional powers

We can use the exact same power rule to differentiate terms with negative or fractional powers.

Make sure to remember that, for negative powers, decreasing the power by 1 means making it more negative (e.g. from -2 to -3).

Example: Differentiate m(x) = 2x^{-3} + 4x^{1/2} - 5

Differentiate each term separately:
- 2x^{-3}: Multiply by -3 and decrease power by 1: 2 \times -3 \times x^{-3-1} = -6x^{-4}.
- 4x^{1/2}: Multiply by 1/2 and decrease power by 1: 4 \times \frac{1}{2} \times x^{\frac{1}{2}-1} = 2x^{-\frac{1}{2}}.
- -5: Constant term, derivative is 0.
Add together the results:
- m'(x) = -6x^{-4} + 2x^{-\frac{1}{2}}.
Answer: m'(x) = -6x^{-4} + 2x^{-\frac{1}{2}}.

Example: Differentiate n(x) = 3x^{3/2} - 2x^{-1} + 7

Differentiate each term separately:
- 3x^{3/2}: Multiply by 3/2 and decrease power by 1: 3 \times \frac{3}{2} \times x^{\frac{3}{2}-1} = \frac{9}{2}x^{\frac{1}{2}}.
- -2x^{-1}: Multiply by -1 and decrease power by 1: -2 \times -1 \times x^{-1-1} = 2x^{-2}.
- 7: Constant term, derivative is 0.
Add together the results:
- n'(x) = \frac{9}{2}x^{\frac{1}{2}} + 2x^{-2}.
Answer: n'(x) = \frac{9}{2}x^{\frac{1}{2}} + 2x^{-2}.

flashcards

Question	Answer
What is the power rule for differentiation?	To differentiate ax^n, multiply by the power n then decrease the power by 1: derivative is a \times n \times x^{n-1}.
How do you differentiate 3x^4?	Multiply by 4, decrease power by 1: 3 \times 4 \times x^{4-1} = 12x^3.
How do you differentiate 2x^3?	Multiply by 3, decrease power by 1: 2 \times 3 \times x^{3-1} = 6x^2.
How do you differentiate -5x^2?	Multiply by 2, decrease power by 1: -5 \times 2 \times x^{2-1} = -10x.
How do you differentiate 7x?	Multiply by 1, decrease power by 1: 7 \times 1 \times x^{1-1} = 7.
What is the derivative of a constant term like -4?	The derivative is 0.
What is the derivative of f(x) = 3x^4 + 2x^3 - 5x^2 + 7x - 4?	f'(x) = 12x^3 + 6x^2 - 10x + 7.
What is the derivative of g(x) = 5x^5 - 3x^4 + x^2 - 8?	g'(x) = 25x^4 - 12x^3 + 2x.
What is the derivative of h(x) = 4x^3 + 6x - 9?	h'(x) = 12x^2 + 6.
What is the derivative of k(x) = 7x^6 - 2x^3 + 5x^2 - x + 1?	k'(x) = 42x^5 - 6x^2 + 10x - 1.
How do you differentiate a term with a negative power, like 2x^{-3}?	Multiply by the negative power and decrease the power by 1 (making it more negative): 2 \times -3 \times x^{-3-1} = -6x^{-4}.
How do you differentiate a term with a fractional power, like 4x^{1/2}?	Multiply by the fraction and decrease the power by 1: 4 \times \frac{1}{2} \times x^{\frac{1}{2}-1} = 2x^{-\frac{1}{2}}.
What is the derivative of m(x) = 2x^{-3} + 4x^{1/2} - 5?	m'(x) = -6x^{-4} + 2x^{-\frac{1}{2}}.
What is the derivative of n(x) = 3x^{3/2} - 2x^{-1} + 7?	n'(x) = \frac{9}{2}x^{\frac{1}{2}} + 2x^{-2}.
What must you remember when decreasing a negative power by 1?	Decreasing a negative power by 1 means making it more negative (e.g., from -2 to -3).