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7 Physics

Arguably the most important application of calculus to physics is the concept of the "time derivative" — the rate of change over time — which is required for the precise definition of several important concepts. In particular, the time derivatives of an object's position are significant in Newtonian physics:

• Velocity (instantaneous velocity; the concept of average velocity predates calculus) is the derivative (with respect to time) of an object's position.
• Acceleration is the derivative (with respect to time) of an object's velocity.
• Jerk is the derivative (with respect to time) of an object's acceleration.

For example, if an object's position ; then, the object's velocity is ; the object's acceleration is ; and the object's jerk is .

If the velocity of a car is given, as a function of time; then, the derivative of said function with respect to time describes the acceleration of said car, as a function of time.

8 Algebraic manipulation

Messy limit calculations can be avoided, in certain cases, because of differentiation rules which allow one to find derivatives via algebraic manipulation; rather than by direct application of Newton's difference quotient. One should not infer that the definition of derivatives, in terms of limits, is unnecessary. Rather, that definition is the means of proving the following "powerful differentiation rules"; these rules are derived from the difference quotient.

• Constant Rule: The derivative of any constant is zero.
  • Constant Multiple Rule: If c is some real number; then, the derivative of equals c multiplied by the derivative of f(x) (a consequence of linearity below)
• Linearity: (af + bg)' = af ' + bg' for all functions f and g and all real numbers a and b.
• General Power Rule (Polynomial rule): If , for some real number r; .
• Product Rule: (fg)' = f 'g + fg' for all functions f and g.
• Quotient Rule: (f/g)' = (f 'g - fg')/ unless g is zero.
• Chain Rule: If f(x) = h(g(x)), then f '(x) = h'[g(x)] * g'(x).
• Inverse functions and differentiation: If , , and f(x) and its inverse are differentiable, then for cases in which when ,
• Derivative of one variable with respect to another when both are functions of a third variable : Let and . Now
• Implicit differentiation: If is an implicit function, we have: dy/dx = - (∂f / ∂x) / (∂f / ∂y).

In addition, the derivatives of some common functions are useful to know. See the table of derivatives.

As an example, the derivative of

is

.

9 Using derivatives to graph functions

Derivatives are a useful tool for examining the graphs of functions. In particular, the points in the interior of the domain of a real-valued function which take that function to local extrema will all have a first derivative of zero. However, not all critical points are local extrema; for example, f(x)=x³ has a critical point at x=0, but it has neither a maximum nor a minimum there. The first derivative test and the second derivative test provide ways to determine if the critical points are maxima, minima or neither.

In the case of multidimensional domains, the function will have a partial derivative of zero with respect to each dimension at local extrema. In this case, the Second Derivative Test can still be used to characterize critical points, by considering the eigenvalues of the Hessian matrix of second partial derivatives of the function at the critical point. If all of the eigenvalues are positive, then the point is a local minimum; if all are negative, it is a local maximum. If there are some positive and some negative eigenvalues, then the critical point is a saddle point, and if none of these cases hold then the test is inconclusive (e.g., eigenvalues of 0 and 3).

Once the local extrema have been found, it is usually rather easy to get a rough idea of the general graph of the function, since (in the single-dimensional domain case) it will be uniformly increasing or decreasing except at critical points, and hence (assuming it is continuous) will have values in between its values at the critical points on either side.

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