Up to now I have been using the prime notation for derivatives.
This notation is simple, and works fine as long as the variable we are taking the derivative with respect to is clear from context.
The situation gets a little more complicated when we have expressions that contain other symbols that stand for symbolic constants.
Sometimes there is no function expression available to give us a hint about which symbol is the variable and which is the constant:
The situation can get really murky in cases like this:
To help make more clear which symbol in an expression is the variable and which symbols are constants, the mathematician Leibniz invented a notation for differentiation.
In situations where the prime notation may not be sufficiently clear we will use the Leibniz notation as a more detailed alternative.
When we talk about functions, most often we specify those functions with an explicit form such as
It is also possible to describe functions implicitly by burying the function in an equation. Perhaps the most common example of this kind of thing is one of the standard forms for an equation of a line. The equation
3 x + 4 y = 1
describes a line. We can solve this equation for y and write y explicitly as a function of x:
We know how to compute the derivatives of functions given in explicit form. In the example above, if we wanted to compute
we would compute
We could compute the same result starting from the implicit description of y. We take the equation which gives y implicitly as a function of x
3 x + 4 y = 1 | (1) |
and differentiate both sides with respect to x.
Just as the equation (1) described y implicitly as a function of x, this equation that we get by differentiating describes the derivative implicitly. In this example, it is easy to solve the resulting equation for the derivative.
Notice that we get the same result by both methods.
As an example of how implicit differentiation can make it easier to solve differentiation problems, consider the problem of computing the equation of the line tangent to an ellipse given by the equation
The key thing needed to compute a tangent line is the slope of the tangent line, which comes from the derivative. One approach is to compute the explicit form of the equation and then differentiate.
The alternative is to leave the equation in implicit form and use implicit differentiation.
We can easily solve this expression for the derivative.
Earlier we proved that if n is an integer
With implicit differentiation we can now prove that
in the case where a = p/q, a rational number. The proof uses the chain rule and implicit differentiation.
In fact, we can prove the derivative formula for irrational a by using a different trick.
Each of the trig function has a corresponding inverse function. For example, the inverse function to sin x is written sin-1 x.
y = sin x
One rather unfortunate and confusing thing about the notation for the inverse trig function is that the notation collides with that for exponentiating trig functions. Thus,
while
If you need to raise a trig function to a negative power, you should write it this way:
In fact, it might be a good thing to abandon the standard exponentiation notation for trig functions altogether and just write
all the time.
To compute the derivative of the inverse sine function we start from the implicit equation
sin y = x
and differentiate both sides with respect to x. By the chain rule we have
This result would be a little more useful if we could write the right hand side as a function of x. To do this, we use the formula
or
The only tricky question left is whether to use the + or - with the radical. If we arbitrarily restrict y to lie in the range 
, then cos y will always be positive and we should use the + sign with the radical.
By similar methods you can compute the derivatives of the other inverse trig functions.
Section 3.6: 3, 4, 13, 16, 25, 26, 42, 43, 54