IV: More Derivatives 😒
Derivatives of Inverse Functions
Derivatives of Exponential and
Logarithmic Functions
inverse function slope relationship
y=arctan x
y=arcsec x
y = arcsin x
dy/dx = 1/ cos y
dy/dx=1/(1+x^2)
dy/dx=1/sec y*tan 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 😃)
d(e^x)/dx=e^x
(Just remains the same ✅)
d(ln x)/dx=1/x
Proof
(chain rule)
For a > 0 and a ≠ 1
=1/(x*lna)
Proof
Power Rule for Arbitrary Real Powers
Proof
For a > 0 and a ≠ 1
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
Look similar
(Chain rule)
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 😃