Differentiation: Composite, Implicit, and Inverse Functions
Six topics on differentiating the functions that are not handed to you in a convenient form. The chain rule and the inner factor everyone drops, implicit differentiation where every y-term carries a dy/dx, the reciprocal slopes of an inverse and the input they are evaluated at, the six inverse trigonometric forms in three mirrored pairs, the discipline of choosing a procedure before applying one, and higher-order derivatives, where every rule has to be applied again.
AB exam 5-10%BC exam 5-10%6 topics
Topics
Key forms For every problem in this unit
Chain rule
d/dx f(g(x)) = f′(g(x)) · g′(x)
Leibniz form
dy/dx = (dy/du) · (du/dx)
Power with an inside
d/dx [un] = n un−1 · u′
Implicit: any y-term
d/dx [y2] = 2y · dy/dx
Implicit: mixed term
d/dx [xy] = y + x · dy/dx
Second derivative
differentiate y′ again, then substitute y′ back in
Inverse function
(f−1)′(x) = 1 / f′(f−1(x))
arcsin x
1 / √(1 − x2)
arccos x
−1 / √(1 − x2)
arctan x
1 / (1 + x2)
arccot x
−1 / (1 + x2)
arcsec x
1 / (|x| √(x2 − 1))
With an inside u
every form above gains a factor of u′ on top