Implicit differentiation and inverse trigonometric functions practice question 5
In implicit differentiation, stands for an unknown function of . Which expression equals ?
Think of as . Then is an outer cube applied to an inner function.
Check the choices with a function you know: if , then , whose derivative is . Which choice gives when you put in ?
Theory UBC Math 100 §4.5CLP-1 §2.11Key results
Worked solution
Strategy
y stands for a function y(x), so y³ is a composition: a cube applied to an inner function. The chain rule differentiates it, and the inner derivative is dy/dx.
Working
Outer function u³, inner function y(x): the outer derivative at the inside, times the inner derivative.
The same result in Leibniz notation.
A check with a known function: both routes agree.
Answer
The answer is 3 y squared times dy/dx.
Check
If y were a constant such as 2, y³ = 8 would have derivative 0, and 3y² dy/dx gives 0, but 3y² alone gives 12.
Where this goes wrong
Writing 3y² and leaving out dy/dx, as if y were the variable. The derivative is taken with respect to x, so every power of y picks up the chain-rule factor dy/dx.
The idea behind it
The implicit differentiation and inverse trigonometric functions lesson, a figure to move, in a few steps
Leans on
The chain rule, from The chain rule and logarithms
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