Implicit Differentiation

In implicit differentiation, differentiate each side of an equation with two variables (usually x  and y by treating one of the variables as a function of the other. Usually, the chain rule is needed.

x² + y²=a² ( constant, a ∈ ℝ)

\(\frac{d}{dx}\)(x² + y²)=\(\frac{d}{dx}\)(a²)

\(\frac{d}{dx}\)(x²) +  \(\frac{d}{dx}\)(y²) = 0

2x + 2y \(\frac{dy}{dx}\)= 0

\(\frac{dy}{dx}\) = – \(\frac{x}{y}\)

Parametric derivatives

If y = f(t)  and x= f(t)

Slope of a tangent line = \(\frac{dy}{dx}\) = \(\frac{dy/dt}{dx/dt}\)

Second derivative = \(\frac{d^2y}{dx^2}\) = \(\frac{\frac{d(dy/dx)}{dt}}{dx/dt}\)  = \(\frac{(dx/dt d^2y/dt^2) - (dy/dt~d^2x/dt^2)}{(dy/dt)^3}\)

Higher Order Derivatives (Velocity Acceleration)

• Second Order derivative : \(\frac{d^2y}{dx^2}\) , f’’(x), f⁽²⁾(x)
• Third Order derivative :  \(\frac{d^3y}{dx^3}\)  , f’’’(x), f⁽³⁾(x)
• n^th -Order derivative : \(\frac{d^ny}{dx^n}\) , fⁿ(x)

                                       Derivative form           Integral form

Position                                   r(t)                        r(t)=r₀+ \(\int_{0}^t vdt\)

Velocity                         v(t)= \(\frac{dr}{dt}\)                   v(t)=v₀+ \(\int_{0}^t vdt\)

Acceleration                       a(t)=\(\frac{dv}{dt}\)= \(\frac{d^2r}{dt^2}\)                a(t)

Critical Points

Let f(x) be a function and let c be a point in the domain of definition of the function. The point c is called a critical point of f(x) if either f′(c)= 0 or f′(c) does not exist.

Increasing Decreasing Interval

The intervals where a function is increasing (or decreasing) correspond to the intervals where its derivative is positive (or negative) that is f’(x)>0  or f’(x)< 0.

Maximum and Minimum Values

A high point is called a maximum.

A low point is called a minimum .

The general word for maximum or minimum is extremum .

We say local maximum (or minimum) when there may be higher (or lower) points elsewhere but not nearby.

Speeding up or Slowing down

An object is speeding up means that its speed is increasing (acceleration > 0) or dot product of velocity and acceleration > 0 ;  slowing down means that its speed is decreasing (acceleration < 0) or dot product of velocity and acceleration < 0 .