Differentiation

Differentiation

You can always use the formal or alternate definition of a derivative to find a functions derivative, but some types of functions have easier methods.

Explicit (normal) differentiation

Constant

Always 0

Linear

In a linear function y = mx + b the derivative will just be m.

Constant multiples

d/dx[cf(x)] = cf'(x)

Find the derivative of the function and multiply by the same constant

Sums and differences

d/dx[f(x) + g(x)] = f'(x) + g'(x)
d/dx[f(x) - g(x)] = f'(x) - g'(x)

Power functions

d/dx(x^n) = nx^(n-1)

Keep definitions in mind for these power functions. If you have 1/2^x you can change to 2^-x. If you have sqrt(x) you actually have x^(1/2).

Also, the derivative of e^x is e^x (https://youtu.be/m2MIpDrF7Es)

Trig functions

d/dx[sin(x)] = cos(x) 😮
d/dx[cos(x)] = -(sin(x))
tan(x)' = sec^2(x)
sec(x)' = sec(x)tan(x)
csc(x)' = -csc(x)cot(x)
cot(x)' = csc^2(x)

Inverse trig functions

It's helpful to review the rules and notation of inverse trig relations and functions.

Proving these definitions is pretty involved so I'm not going to put it here.

arctan(x) -> 1/(1+x^2)
arcsin(x) -> 1/sqrt(1-x^2)

ln(x)

d/dx[ln(x)] = 1/x

Product rule

d/dx[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)

Quotient rule

with f(x) = u(x) / v(x)

f'(x) = [u'(x)v(x) - u(x)v'(x)] / [(v(x))^2]

The Chain Rule (compositions of functions)

You can use the chain rule to find derivatives for compositions of functions

Suppose a(x) = (f°g)(x) = f(g(x)) where f and g are both differentiable
functions. then

a'(x) = f'(g(x))g'(x)

Example: (x^2 + 1)^10

(x^2 + 1)^10 can be rewritten as a composite function f(g(x))

f(u) = u^10
g(x) = x^2 + 1

So therefore

f'(u) = 10u^9
g'(x) = 2x

Written in chain rule form

(10(x^2 + 1)^9) * 2x

Natural Logarithm and Exponential

Keep in mind the properties of logarithms

ln'(x) = 1/x

ln'(x^2 + 1) -> use chain rule -> (1/(x^2+1))(2x + 0)

a^x -> (a^x)ln(a)

Implicit differentiation

If you have a statement not written as a function of x, like x^2 + y^2 = 16 you have to use implicit differentiation.

  1. Differentiate both sides with respect to x. When you differentiate a y term, multiply by dy/dx.
  2. Solve for dy/dx.
(x^2 + y^2 = 36)'
2x + 2y(dy/dx) = 0
2y(dy/dx) = -2x
dy/dx = -2x/2y
dy/dx = -x/y
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