Please enable JavaScript.
Coggle requires JavaScript to display documents.
IV: More Derivatives :unamused:, Look similar (Chain rule) - Coggle…
IV: More Derivatives :unamused:
Derivatives of Inverse Functions
inverse function slope relationship
y=arctan x
dy/dx=1/(1+x^2)
y=arcsec x
dy/dx=1/sec y*tan y
y = arcsin x
dy/dx = 1/ cos y
∵
d(arccos x)/dx =-d(arcsin x)/dx
∴ d(arccot x)/dx= -d(arctan x)/dx
d(arccsc x)/dx= -d(arcsec x)/dx
(just add a negative sign before the 3 above :smiley:)
Derivatives of Exponential and
Logarithmic Functions
d(e^x)/dx=e^x
(Just remains the same :check:)
Proof
（chain rule）
For a ＞ 0 and a ≠ 1
d(ln x)/dx=1/x
Proof
=1/(x*lna)
Proof
For a ＞ 0 and a ≠ 1
Power Rule for Arbitrary Real Powers
Proof
implicit function
Differentiate both sides of the equation with respect to x
Collect the terms with dy/dx on one side of the equation.
Factor out dy/dx
Solve for dy/dx.
example
Power Rule for Rational Powers of x
If n is any rational number, then
If n ＜1, then the derivative does not exist at x = 0
Chain rule
If ƒ is differentiable at the point u = g(x), and g is differentiable at x, then
In Leibniz notation, if y = ƒ(u) and u = g(x), then
Similarly,
example: Differentiate sin(x^2 + x)^2 with respect to x.
solution: d(sin(x^2 + x)^2)/dx
= cos (x^2 + x) * (2x + 1)
It's like peel an onion :smiley:
Look similar
(Chain rule)
