Differentiation

Implicit Differentiation

When differentiating a function of one variable with respect to another (ie ), simply differentiate with respect to , then multiply by .

For example, find where . First, using the product rule to differentiate the first term:

The equation with all terms differentiated:

Rearranging to get in terms of :

Inverse Trig Functions

All the derivatives of the inverse trig functions are given in the data book. They can be derived as follows ( is used as an example).

Differentiating both sides with respect to x

Using pythagorean identity

Differentials

Differentials describe small changes to values/functions

Recall that . This means this can be rewritten:

Dividing both sides by :

represents a relative change in y, and represents a relative change in x. This can be used to give approximations of how one quantity changes based upon another.

For example, given the mass of a sphere , where is the material density, estimate the change in mass when the radius is increased by 2%.

Dividing both sides by the original formula:

represents a relative change in radius, so when increases by 2%,

Meaning the mass increases by 6%.

Hyperbolic Functions

Hyperbolic functions have similar identities to circular trig functions. They're the same, except anywhere there is a product of two s, the term should be negated. Hyperbolic functions can also be defined in terms of exponential functions, making them easy to differentiate.

All the derivatives of hyperbolic functions are given in the formula book.

Parametric Differentiation

For a function given in parametric form , :

Partial Differentiation

For a function of two variables there are two gradients at the point , one in and one in . To find the gradient in the x direction, differentiate treating y as a constant. To find the gradient in the y direction, differentiate treating x as a constant. These are the two partial derivatives of the function, and .

For example, for a function :

Implicit Partial Differentiation

When a function of several variables is given and a partial derivative is required, differentiate the numerator of the partial derivative implicitly with respect to the denominator, and treat the third variable as constant. For example, find given :

Another example, find given

Higher Order Partial Derivatives

Three 2nd order derivatives for functions of 2 variables. For :

Note how for the last one, the order is interchangable as it yields the same result.

Chain Rule

The chain rule for a function , where x and y are functions of a parameter :

Total Differential

The total differential represents the total height gain or lost when moving along the function described by

Contour Plots

Along a line of a contour plot, the total differential is zero: the height doesn't change. This allows to be found