IV: More Derivatives 😒

Derivatives of Inverse Functions

Derivatives of Exponential and
Logarithmic Functions

inverse function slope relationship
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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

image

image

∵ image

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 image

（chain rule）
For a ＞ 0 and a ≠ 1
image

image =1/(x*lna)

Proof image

image

Power Rule for Arbitrary Real Powers

Proof image

For a ＞ 0 and a ≠ 1
image

image

Proof image

implicit function

  1. Differentiate both sides of the equation with respect to x
  2. Collect the terms with dy/dx on one side of the equation.
  3. Factor out dy/dx
  4. Solve for dy/dx.

example image

Power Rule for Rational Powers of x

If n is any rational number, then
image
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
image image
In Leibniz notation, if y = ƒ(u) and u = g(x), then
image

Similarly, image

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 😃