VIDEO: Calculate the slope of any function

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Slope of a function - the derivative

  • A linear function (also called a straight line) has the same slope at any point. You can find it in the function equation y = mx + b, namely the value "m".
  • For general or any Functions things look different. Even a quadratic function (parabola) has different slopes in different points - sometimes the function goes steeply up, sometimes steeply down and at the apex it does not rise at all.
  • But the slope can also be calculated for such functions. However, you shouldn't expect numerical values ​​as the gradient, but rather a calculation formula.
  • This is the derivative f '(x) of the function that you learned about in differential calculus.
  • With the derivative you can calculate the slope of the function for any point (the x-value is even sufficient). You need to plug the x-value into the derivative and calculate the term.
    • Read off the slope of parabolas

    Are you currently studying parables? Then you must certainly also ...

  • The prerequisite for this is, of course, that you know the derivation for any function. The formulas (or the Internet) can help here. In addition, the derivative of many functions can be calculated using known derivation rules.

Calculating the slope - an example of the procedure

For the function f (x) = 1 / x you should calculate the slope at the point x = -2 and decide whether the function decreases or increases there.

  1. You know, calculate or look up the derivative of f (x) = 1 / x in a collection of formulas - note for calculators: 1 / x = x-1, then apply the rule for power functions f '(x) = n * xn-1
  2. You get f '(x) = -1 * x-2= -1 / x2.
  3. Now insert x = -2 into this derivative and get the slope f '(- 2) = -1 / (- 2)2 = -1/4. Be sure to properly dissolve the potency.
  4. The slope at point x = -2 is therefore -1/4. The function falls there because the slope is negative.
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