What are first and second order derivatives and how to calculate them?

First and second-order derivatives are the types of derivatives. In calculus, the derivative is the rate of change of the function with respect to its variable. A derivative is used to calculate the slope. Derivatives are widely used in mathematics, physics, and many other branches of science.

In general, the way of determining the derivatives is said to be the differentiation. In derivative the function can be a singleton function or in the form of the equation.

First Order Derivative

In calculus, to determine the direction of the function, the function can be increasing or decreasing, we use the first-order derivative. It is defined as the immediate rate of change in the given function. The first-order derivative is also used to determine the slope of the tangent.

In simple words, the rate of change of inconstant with respect to other inconstant of a function is known as the first-order derivative. For example, in physics, we state that the rate of change of the velocity with respect to a specific interval of time is known as acceleration. In this statement, velocity is the dependent variable and time is the independent variable.

To determine the acceleration, we have to apply the first-order derivative of the velocity. First-order differentiation is the other name of the first-order derivative. Differentiation is used in algebra in the place of derivatives to find differential at any point. While in calculus, the derivative concept is used.

Second Order Derivative

In calculus, to find the derivative of the first derivative already taken from a function. To determine the idea of the shape of the graph, we use a second-order derivative. A second-order derivative is generally used for the term concavity. The concavity is of two types in a graph.

• Concave up
• Concave down

The basic work of the derivative is to find the slope. When the slope is continuously increasing the slope is said to be concave up, on the other hand, when the slope is continuously decreasing the slope is said to be the concave down.

How to evaluate derivative problems?

A derivative is the rate of change of the function, function can be increasing or decreasing, in the respect of the given variable. The derivative can be calculated by using formulas, rules, and trigonometric ratios. The derivative of a function can be calculated by using a derivative calculator which is free of cost and gives the accurate output of the given input.

Example 1

Evaluate the first and second-order derivative of the given function, (3x + 9) * (4x – 2x³) with respect to x.

Solution

Step 1: Determine the given function.

F(x) = (3x + 9) * (4x – 2x³)

Step 2: Write the formula of derivative.

d/dx (f(x))

Step 3: Place the value of the function in the formula.

d/dx ((3x + 9) * (4x – 2x³))

Step 4: Now apply product law of the functions.

d/dx ((3x + 9) * (4x – 2x³)) = (3x + 9) d/dx (4x – 2x³) + (4x – 2x³) d/dx (3x +9)

Step 5: Now apply the laws of derivatives such as sum, difference, power, and constant.

d/dx ((3x + 9) * (4x – 2x³)) = (3x + 9) d/dx (4x – 2x³) + (4x – 2x³) d/dx (3x +9)

= (3x + 9) (d/dx (4x) – d/dx (2x³)) + (4x – 2x³) (d/dx (3x) + d/dx (9))

= (3x + 9) ((4x^1-1) – (2×3 x^3-1)) + (4x – 2x³) ((3x^1-1) + 0)

= (3x + 9) (4 – 6x²) + (3x – 4x²) (3 + 0)

= (3x + 9) (4 – 6x²) + (4x – 2x³) (3)

= 12x – 18x³ + 36 – 54x² + 12x – 6x³

= 24x – 54x² – 24x³ + 36

d/dx ((3x + 9) * (4x – 2x³)) = -6 (-4x + 9x² + 4x³ – 6)

Second order derivative

For second order derivative, we take the derivative of the first derivative.

Step 1: Take the derivative of the first derivative.

d/dx (d/dx ((3x + 9) * (4x – 2x³))) =d/dx (-6 (-4x + 9x² + 4x³ – 6))

d²/dx² ((3x + 9) * (4x – 2x³)) =d/dx (-6 (-4x + 9x² + 4x³ – 6))

d²/dx² ((3x + 9) * (4x – 2x³)) =d/dx (24x – 54x² – 24x³ + 36)

Step 2: Apply the laws of derivatives such as sum, difference, power, and constant.

d²/dx² ((3x + 9) * (4x – 2x³)) = d/dx (24x) – d/dx (54x²) – d/dx (24x³) + d/dx (36)

= 24x^1-1 – (54 x 2) x^2-1 – (24 x 3) x^3-1 + 0

= 24 – 108x – 72x²

Hence, the first and second derivatives of the functions are calculated.

Example 2

Evaluate the first and second order derivative of the given function, (2x + 9x²) / (x) with respect to x.

Solution

Step 1: Determine the given function.

F(x) = (2x + 9x²) / (x)

Step 2: Write the formula of derivative.

d/dx (f(x))

Step 3: Place the value of the function in the formula.

d/dx ((2x + 9x²) / (x))

Step 4: Now apply quotient law of the functions.

d/dx ((2x + 9x²) / (x)) = 1/x² [(x d/dx (2x – 9x²) – (2x + 9x²) d/dx (x)]

Step 5: Now apply the laws of derivatives such as sum, difference, power, and constant.

d/dx ((2x + 9x²) / (x)) = 1/x² [(x d/dx (2x – 9x²) – (2x + 9x²) d/dx (x)]

= 1/x² [(x) (d/dx (2x) – d/dx (9x²)) – (2x + 9x²) d/dx (x)]

= 1/x² [(x) ((2x^1-1) – (9 x 2) x^2-1)) – (2x + 9x²) (x^1-1)]

= 1/x² [(x) ((2) – (18) x)) – (2x + 9x²) (x⁰)]

= 1/x² [(x) (2 – 18x) – (2x + 9x²) (1)]

= 1/x² [2x – 18x²) – (2x + 9x²)]

= 1/x² [2x – 18x² – 2x – 9x²]

= 1/x² [27x²]

d/dx ((2x + 9x²) / (x)) = 27

Second order derivative

For second order derivative, we take the derivative of the first derivative.

Step 1: Take the derivative of the first derivative.

d/dx (d/dx ((2x + 9x²) / (x))) =d/dx (27)

d²/dx² ((2x + 9x²) / (x)) = d/dx (27)

Step 2: Apply the laws of derivatives such as sum, difference, power, and constant.

d²/dx² ((2x + 9x²) / (x)) = d/dx (27)

d²/dx² ((2x + 9x²) / (x)) = 0

Hence, the first and second derivatives of the functions are calculated.

Summary

First and second-order derivatives are the types of derivatives. All the problems are easily solved by applying the rules and formulas of the derivatives.