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For the tangent function, see .
For other uses, see .
Tangent to a curve. The red line is tangential to the curve at the point marked by a red dot.
Tangent plane to a sphere
In , the tangent line (or simply tangent) to a plane
at a given
that "just touches" the curve at that point.
defined it as the line through a pair of
points on the curve. More precisely, a straight line is said to be a tangent of a curve y = f(x) at a point x = c on the curve if the line passes through the point (c, f(c)) on the curve and has slope f '(c) where f ' is the
of f. A similar definition applies to
and curves in n-dimensional .
As it passes through the point where the tangent line and the curve meet, called the point of tangency, the tangent line is "going in the same direction" as the curve, and is thus the best straight-line approximation to the curve at that point.
Similarly, the tangent plane to a
at a given point is the
that "just touches" the surface at that point. The concept of a tangent is one of the most fundamental notions in
and has been ext see .
The word tangent comes from the
, to touch.
The first definition of a tangent was "a right line which touches a curve, but which when produced, does not cut it". This old definition prevents
from having any tangent. It has been dismissed and the modern definitions are equivalent to those of . The tangent problem has given rise to . The main ideas behind
are due to
and were developed by , ,
developed a general technique for determining the tangents of a curve using his method of
in the 1630s.
Leibniz defined the tangent line as the line through a pair of
points on the curve.
A tangent, a , and a
to a circle
The intuitive notion that a tangent line "touches" a curve can be made more explicit by considering the sequence of straight lines () passing through two points, A and B, those that lie on the function curve. The tangent at A is the limit when point B approximates or tends to A. The existence and uniqueness of the tangent line depends on a certain type of mathematical smoothness, known as "differentiability." For example, if two circular arcs meet at a sharp point (a vertex) then there is no uniquely defined tangent at the vertex because the limit of the progression of secant lines depends on the direction in which "point B" approaches the vertex.
At most points, the tangent touches the curve without crossing it (though it may, when continued, cross the curve at other places away from the point of tangent). A point where the tangent (at this point) crosses the curve is called an . , ,
do not have any inflection point, but more complicated curves do have, like the graph of a , which has exactly one inflection point.
Conversely, it may happen that the curve lies entirely on one side of a straight line passing through a point on it, and yet this straight line is not a tangent line. This is the case, for example, for a line passing through the vertex of a
and not intersecting the triangle—where the tangent line does not exist for the reasons explained above. In , such lines are called .
At each point, the line is always tangent to the . I the positivity, negativity and zeroes of the derivative are marked by green, red and black respectively.
The geometrical idea of the tangent line as the limit of secant lines serves as the motivation for analytical methods that are used to find tangent lines explicitly. The question of finding the tangent line to a graph, or the tangent line problem, was one of the central questions leading to the development of
in the 17th century. In the second book of his ,
of the problem of constructing the tangent to a curve, "And I dare say that this is not only the most useful and most general problem in geometry that I know, but even that I have ever desired to know".
Suppose that a curve is given as the graph of a , y = f(x). To find the tangent line at the point p = (a, f(a)), consider another nearby point q = (a + h, f(a + h)) on the curve. The
passing through p and q is equal to the
As the point q approaches p, which corresponds to making h smaller and smaller, the difference quotient should approach a certain limiting value k, which is the slope of the tangent line at the point p. If k is known, the equation of the tangent line can be found in the point-slope form:
To make the preceding reasoning rigorous, one has to explain what is meant by the difference quotient approaching a certain limiting value k. The precise mathematical formulation was given by
in the 19th century and is based on the notion of . Suppose that the graph does not have a break or a sharp edge at p and it is neither plumb nor too wiggly near p. Then there is a unique value of k such that, as h approaches 0, the difference quotient gets closer and closer to k, and the distance between them becomes negligible compared with the size of h, if h is small enough. This leads to the definition of the slope of the tangent line to the graph as the limit of the difference quotients for the function f. This limit is the
of the function f at x = a, denoted f ′(a). Using derivatives, the equation of the tangent line can be stated as follows:
Calculus provides rules for computing the derivatives of functions that are given by formulas, such as the , , , , and their various combinations. Thus, equations of the tangents to graphs of all these functions, as well as many others, can be found by the methods of calculus.
Calculus also demonstrates that there are functions and points on their graphs for which the limit determining the slope of the tangent line does not exist. For these points the function f is non-differentiable. There are two possible reasons for the method of finding the tangents based on the limits and derivatives to fail: either the geometric tangent exists, but it is a vertical line, which cannot be given in the point-slope form since it does not have a slope, or the graph exhibits one of three behaviors that precludes a geometric tangent.
The graph y = x1/3 illustrates the first possibility: here the difference quotient at a = 0 is equal to h1/3/h = h-2/3, which becomes very large as h approaches 0. This curve has a tangent line at the origin that is vertical.
The graph y = x2/3 illustrates another possibility: this graph has a
at the origin. This means that, when h approaches 0, the difference quotient at a = 0 approaches plus or minus infinity depending on the sign of x. Thus both branches of the curve are near to the half vertical line for which y=0, but none is near to the negative part of this line. Basically, there is no tangent at the origin in this case, but in some context one may consider this line as a tangent, and even, in , as a double tangent.
The graph y = |x| of the
function consists of two straight lines with different slopes joined at the origin. As a point q approaches the origin from the right, the secant line always has slope 1. As a point q approaches the origin from the left, the secant line always has slope -1. Therefore, there is no unique tangent to the graph at the origin. Having two different (but finite) slopes is called a corner.
Finally, since differentiability implies continuity, the
states discontinuity implies non-differentiability. Any such jump or point discontinuity will have no tangent line. This includes cases where one slope approaches positive infinity while the other approaches negative infinity, leading to an infinite jump discontinuity
When the curve is given by y = f(x) then the slope of the tangent is
the equation of the tangent line at (X, Y) is
where (x, y) are the coordinates of any point on the tangent line, and where the derivative is evaluated at .
When the curve is given by y = f(x), the tangent line's equation can also be found by using
by ; if the remainder is denoted by , then the equation of the tangent line is given by
When the equation of the curve is given in the form f(x, y) = 0 then the value of the slope can be found by , giving
The equation of the tangent line at a point (X,Y) such that f(X,Y) = 0 is then
This equation remains true if
(in this case the slope of the tangent is infinite). If
the tangent line is not defined and the point (X,Y) is said .
For , computations may be simplified somewhat by converting to . Specifically, let the homogeneous equation of the curve be g(x, y, z) = 0 where g is a homogeneous function of degree n. Then, if (X, Y, Z) lies on the curve,
It follows that the homogeneous equation of the tangent line is
The equation of the tangent line in Cartesian coordinates can be found by setting z=1 in this equation.
To apply this to algebraic curves, write f(x, y) as
where each ur is the sum of all terms of degree r. The homogeneous equation of the curve is then
Applying the equation above and setting z=1 produces
as the equation of the tangent line. The equation in this form is often simpler to use in practice since no further simplification is needed after it is applied.
If the curve is given
then the slope of the tangent is
giving the equation for the tangent line at
If , the tangent line is not defined. However, it may occur that the tangent line exists and may be computed from an implicit equation of the curve.
The line perpendicular to the tangent line to a curve at the point of tangency is called the normal line to the curve at that point. The slopes of perpendicular lines have product -1, so if the equation of the curve is y = f(x) then slope of the normal line is
and it follows that the equation of the normal line at (X, Y) is
Similarly, if the equation of the curve has the form f(x, y) = 0 then the equation of the normal line is given by
If the curve is given parametrically by
then the equation of the normal line is
The angle between two curves at a point where they intersect is defined as the angle between their tangent lines at that point. More specifically, two curves are said to be tangent at a point if they have the same tangent at a point, and orthogonal if their tangent lines are orthogonal.
The lima?on trisectrix: a curve with two tangents at the origin.
The formulas above fail when the point is a . In this case there may be two or more branches of the curve which pass through the point, each branch having its own tangent line. When the point is the origin, the equations of these lines can be found for algebraic curves by factoring the equation formed by eliminating all but the lowest degree terms from the original equation. Since any point can be made the origin by a change of variables, this gives a method for finding the tangent lines at any singular point.
For example, the equation of the
shown to the right is
Expanding this and eliminating all but terms of degree 2 gives
which, when factored, becomes
So these are the equations of the two tangent lines through the origin.
Two pairs of tangent circles. Above internally and below externally tangent
Two circles of non-equal radius, both in the same plane, are said to be tangent to each other if they meet at only one point. Equivalently, two , with
of ri and centers at (xi, yi), for i = 1, 2 are said to be tangent to each other if
Two circles are externally tangent if the
between their centres is equal to the sum of their radii.
Two circles are internally tangent if the
between their centres is equal to the difference between their radii.
Main article:
The tangent plane to a
at a given point p is defined in an analogous way to the tangent line in the case of curves. It is the best approximation of the surface by a plane at p, and can be obtained as the limiting position of the planes passing through 3 distinct points on the surface close to p as these points converge to p. More generally, there is a k-dimensional
at each point of a k-dimensional
in the n-dimensional .
Leibniz, G., Nova methodus pro maximis et minimis ..., Acta Erud., Oct. 1684
Noah Webster, American Dictionary of the English Language (New York: S. Converse, 1828), vol. 2, p.733,
Descartes, René (1954). The geometry of René Descartes. Courier . p. 95.  .
R. E. Langer (October 1937). "Rene Descartes".
(Mathematical Association of America) 44 (8): 495–512. :.  .
Edwards Art. 191
Strickland-Constable, Charles, "A simple method for finding tangents to polynomial graphs", , November –467.
Edwards Art. 192
Edwards Art. 193
Edwards Art. 196
Edwards Art. 194
Edwards Art. 195
Edwards Art. 197
J. Edwards (1892). . London: MacMillan and Co. pp. 143 ff.
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has the text of the 1921
Hazewinkel, Michiel, ed. (2001), , , ,  
With interactive animation
— An interactive simulation
: Hidden categories:Implicit Differentiation
IMPLICIT DIFFERENTIATION PROBLEMS
The following problems require the use of implicit differentiation. Implicit differentiation is nothing more than a special case of the well-known chain rule for derivatives.
The majority of differentiation problems in first-year calculus involve functions y written EXPLICITLY as functions of x .
For example,
then the derivative of y is
However, some functions y are written IMPLICITLY as functions of x .
A familiar example of this is the equation
x2 + y2 = 25 ,
which represents a circle of radius five centered at the origin.
Suppose that we wish to find the slope of the line tangent to the graph of this equation at the point (3, -4) .
How could we find the derivative of y in this instance ?
One way is to first write y explicitly as a function of x .
x2 + y2 = 25 ,
y2 = 25 - x2 ,
where the positive square root represents the top semi-circle and the negative square root represents the bottom semi-circle.
Since the point (3, -4) lies on the bottom semi-circle given by
the derivative of y is
Thus, the slope of the line tangent to the graph at the point (3, -4) is
Unfortunately, not every equation involving x and y can be solved explicitly for y .
For the sake of illustration we will find the derivative of y WITHOUT writing y explicitly as a function of x .
Recall that the derivative (D) of a function of x squared, (f(x))2 , can be found using the chain rule :
Since y symbolically represents a function of x, the derivative of y2 can be found in the same fashion :
Now begin with
x2 + y2 = 25 .
Differentiate both sides of the equation, getting
D ( x2 + y2 ) = D ( 25 ) ,
D ( x2 ) + D ( y2 ) = D ( 25 ) ,
2x + 2 y y' =
Thus, the slope of the line tangent to the graph at the point (3, -4) is
This second method illustrates the process of implicit differentiation.
It is important to note that the derivative expression for explicit differentiation involves x only, while the derivative expression for implicit differentiation may involve BOTH x AND y .
The following problems range in difficulty from average to challenging.
Assume that y is a function of x .
y' = dy/dx for
x3 + y3 = 4 .
Click to see a detailed solution to problem 1.
Assume that y is a function of x .
y' = dy/dx for
(x-y)2 = x + y - 1 .
Click to see a detailed solution to problem 2.
Assume that y is a function of x .
y' = dy/dx for
Click to see a detailed solution to problem 3.
Assume that y is a function of x .
y' = dy/dx for
y = x2 y3 + x3 y2 .
Click to see a detailed solution to problem 4.
Assume that y is a function of x .
y' = dy/dx for
exy = e4x - e5y .
Click to see a detailed solution to problem 5.
Assume that y is a function of x .
y' = dy/dx for
Click to see a detailed solution to problem 6.
Assume that y is a function of x .
y' = dy/dx for
Click to see a detailed solution to problem 7.
Assume that y is a function of x .
y' = dy/dx for
Click to see a detailed solution to problem 8.
Assume that y is a function of x .
y' = dy/dx for
Click to see a detailed solution to problem 9.
Find an equation of the line tangent to the graph of
(x2+y2)3 = 8x2y2
at the point (-1, 1) .
Click to see a detailed solution to problem 10.
Find an equation of the line tangent to the graph of
x2 + (y-x)3 = 9
Click to see a detailed solution to problem 11.
Find the slope and concavity of the graph of
x2y + y4 = 4 + 2x at the point (-1, 1) .
Click to see a detailed solution to problem 12.
Consider the equation
x2 + xy + y2 = 1 .
Find equations for y' and y'' in terms of x and y only.
Click to see a detailed solution to problem 13.
Find all points (x, y) on the graph of
x2/3 + y2/3 = 8 (See diagram.) where lines tangent to the graph at (x, y) have slope
Click to see a detailed solution to problem 14.
The graph of
x2 - xy + y2 = 3 is a "tilted" ellipse (See diagram.).
Among all points (x, y) on this graph, find the largest and smallest
values of y .
Among all points (x, y) on this graph, find the largest and smallest values of x .
Click to see a detailed solution to problem 15.
Find all points (x, y) on the graph of
(x2+y2)2 = 2x2-2y2 (See diagram.) where y' = 0.
Click to see a detailed solution to problem 16.
to return to the original list of various types of calculus problems.
Your comments and suggestions are welcome.
Please e-mail any correspondence to Duane Kouba by
clicking on the following address :
Duane Kouba}