Why is the derivative of (3x + 1)⁵ not 5(3x + 1)⁴?
Updated
The short answer
When one function sits inside another, the derivative is the derivative of the outside times the derivative of the inside. So the derivative of (3x + 1)⁵ is 5(3x + 1)⁴ · 3 = 15(3x + 1)⁴, and the usual slip is leaving off the 3.
What it looks like
| Function | Wrong derivative | Right derivative |
|---|---|---|
| (3x + 1)⁵ | 5(3x + 1)⁴ | 15(3x + 1)⁴ |
| sin(2x) | cos(2x) | 2 cos(2x) |
| (x² + 1)³ | 3(x² + 1)² | 6x(x² + 1)² |
In each wrong answer the outside part was differentiated correctly. What's missing is the last factor, the derivative of the inside: 3, 2 and 2x.
In alai, an iPad app that reads your handwriting as you work, you can hold to talk and ask the tutor, Ally, about a derivative you're unsure of.
Why it happens
The power rule and the basic derivatives are practiced first on plain x. The derivative of x⁵ is 5x⁴, and the chain rule is still at work there, but the derivative of the inside, x, is 1, so it never shows. Once the inside is 3x + 1, that hidden factor becomes 3, and it has to be written down.
Here is why the factor belongs there. The outside derivative, 5(3x + 1)⁴, tells you how fast the fifth power changes as the inside changes. But the inside, 3x + 1, changes 3 times as fast as x does. So the whole function changes 3 times as fast again, and the derivative is 5(3x + 1)⁴ · 3.
To find the outside, Paul Dawkins suggests asking how you would work out the function for a number. For (3x + 1)⁵ at x = 2, you'd work out 3(2) + 1 = 7 first, then 7⁵. The last thing you do is the outside function, the fifth power, and everything before it is the inside.
How to catch it yourself
Before you write the derivative, name the inside. Once you've named it, the derivative has to end with the derivative of that inside. Then point at the inside of your function and ask yourself:
Where is the derivative of the 3x + 1 in your answer?
You can also test the rule on a function you can expand. (2x)² is 4x², whose derivative is 8x. The chain rule gives 2(2x) · 2 = 8x too, while leaving off the inside gives only 4x.
You need the chain rule whenever something other than plain x sits inside another function, for example inside parentheses with a power, under a root, inside a sine or cosine, or in an exponent.
Try these
Differentiate each one.
- (2x − 5)⁴
- cos(5x)
- √(x² + 1)
Answers
- 8(2x − 5)³, which is 4(2x − 5)³ · 2
- −5 sin(5x)
- x/√(x² + 1), which is 1/(2√(x² + 1)) · 2x
How alai helps
If you're not sure a derivative is right, hold to talk and tell Ally what you got. Ally answers out loud with a hint, such as a question about the derivative of the inside. Ally also remembers what tripped you up in past sessions, so a later session can start from there.