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A closely related concept to the derivative of a function is its differential. When x and y are real variables, the derivative of f at x is the slope of the tangent line to the graph of f at x. Because the source and target of f are one-dimensional, the derivative of f is a real number.
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 derivative of the function f at x = a, denoted f ′(a). Using derivatives, the equation of the tangent line can be stated as follows: = + ′ ().
In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point.
The derivative of the function at a point is the slope of the line tangent to the curve at the point. The slope of the constant function is 0, because the tangent line to the constant function is horizontal and its angle is 0.
As h approaches zero, the slope of the secant line approaches the slope of the tangent line. Therefore, the true derivative of f at x is the limit of the value of the difference quotient as the secant lines get closer and closer to being a tangent line: ′ = (+) ().
For this reason, the derivative is sometimes called the slope of the function f. [48]: 61–63 Here is a particular example, the derivative of the squaring function at the input 3. Let f(x) = x 2 be the squaring function. The derivative f′(x) of a curve at a point is the slope of the line tangent to that curve at that point. This slope is ...
Implicit differentiation gives the formula for the slope of the tangent line to this curve to be [3] =. Using either one of the polar representations above, the area of the interior of the loop is found to be 3 a 2 / 2 {\displaystyle 3a^{2}/2} .
Slope illustrated for y = (3/2)x − 1.Click on to enlarge Slope of a line in coordinates system, from f(x) = −12x + 2 to f(x) = 12x + 2. The slope of a line in the plane containing the x and y axes is generally represented by the letter m, [5] and is defined as the change in the y coordinate divided by the corresponding change in the x coordinate, between two distinct points on the line.