derivative

Examples of derivative in the following topics:

• Higher Derivatives

  • The derivative of an already-differentiated expression is called a higher-order derivative.
  • The second derivative, or second order derivative, is the derivative of the derivative of a function.
  • Furthermore, the third derivative is the derivative of the derivative of the derivative of a function, which can be represented by $f'''(x)$.
  • This can continue as long as the resulting derivative is itself differentiable, with the fourth derivative, the fifth derivative, and so on.
  • Any derivative beyond the first derivative can be referred to as a higher order derivative.

• The Derivative as a Function

  • If every point of a function has a derivative, there is a derivative function sending the point $a$ to the derivative of $f$ at $x = a$: $f'(a)$.
  • Because every point $a$ has a derivative, there is a function that sends the point $a$ to the derivative of $f$ at $a$.
  • This function is written $f'(x)$ and is called the derivative function or the derivative of $f$.
  • The derivative of $f$ collects all the derivatives of $f$ at all the points in the domain of $f$.
  • By the definition of the derivative function, $D(f)(a)=f'(a)$.

• Partial Derivatives

  • A partial derivative of a function of several variables is its derivative with respect to a single variable, with the others held constant.
  • A partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary).
  • Partial derivatives are used in vector calculus and differential geometry.
  • That is to say, the partial derivative of $z$ with respect to $x$ at $(1, 1, 3)$ is $3$.
  • Like ordinary derivatives, the partial derivative is defined as a limit.

• Concavity and the Second Derivative Test

  • The second derivative test is a criterion for determining whether a given critical point is a local maximum or a local minimum.
  • In calculus, the second derivative test is a criterion for determining whether a given critical point of a real function of one variable is a local maximum or a local minimum using the value of the second derivative at the point.
  • Now, by the first derivative test, $f(x)$ has a local minimum at $x$.
  • Telling whether a critical point is a maximum or a minimum has to do with the second derivative.
  • Calculate whether a function has a local maximum or minimum at a critical point using the second derivative test

• Derivatives of Logarithmic Functions

  • The general form of the derivative of a logarithmic function is $\frac{d}{dx}\log_{b}(x) = \frac{1}{xln(b)}$.
  • Here, we will cover derivatives of logarithmic functions.
  • It should be noted that what we want is the derivative of y, or $\frac{dy}{dx}$.
  • Applying the chain rule and the property of exponents we derived earlier, we can take the derivative of both sides:
  • Since $\frac{1}{\ln(b)}$ is a constant, we can take it out of the derivative:

• Derivatives of Exponential Functions

  • The derivative of the exponential function is equal to the value of the function.
  • The importance of the exponential function in mathematics and the sciences stems mainly from properties of its derivative.
  • Graph of the exponential function illustrating that its derivative is equal to the value of the function.
  • Since the slope of the red tangent line (the derivative) at $P$ is equal to the ratio of the triangle's height to the triangle's base (rise over run), and the derivative is equal to the value of the function, $h$ must be equal to the ratio of $h$ to $b$.

• Derivatives of Trigonometric Functions

  • Derivatives of trigonometric functions can be found using the standard derivative formula.
  • With this in mind, we can use the definition of a derivative to calculate the derivatives of different trigonometric functions:
  • The same procedure can be applied to find other derivatives of trigonometric functions.
  • In this image, one can see that where the line tangent to one curve has zero slope (the derivative of that curve is zero), the value of the other function is zero.

• Directional Derivatives and the Gradient Vector

  • The directional derivative represents the instantaneous rate of change of the function, moving through $\mathbf{x}$ with a velocity specified by $\mathbf{v}$.
  • It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the coordinate curves, all other coordinates being constant.
  • The name "directional derivative" is a bit misleading since it depends on both the length and direction of $\mathbf v$.
  • Many of the familiar properties of the ordinary derivative hold for the directional derivative.
  • Directional derivative represents the rate of change of the function along any direction specified by $\mathbf{v}$.

• Differentiation and Rates of Change in the Natural and Social Sciences

  • This rate of change is called the derivative of $y$ with respect to $x$.
  • For example, in physics, the derivative of the displacement of a moving body with respect to time is the velocity of the body, and the derivative of velocity with respect to time is acceleration.
  • The reaction rate of a chemical reaction is a derivative.
  • In operations research, derivatives determine the most efficient ways to transport materials and design factories.
  • Derivatives are frequently used to find the maxima and minima of a function.

• Derivatives and the Shape of the Graph

  • The shape of a graph may be found by taking derivatives to tell you the slope and concavity.
  • This rate of change is called the derivative of $y$ with respect to $x$.
  • A point where the second derivative of a function changes sign is called an inflection point.
  • At each point, the derivative of is the slope of a line that is tangent to the curve.
  • The line is always tangent to the blue curve; its slope is the derivative.