In calculus, the most common concept is taking integrals and derivatives. A derivative is the algebraic manipulation of an expression to find the instantaneous rate of change.
This manipulation is referred to as the first derivative when done on the original expression.
However, in calculus, you can also manipulate derivatives to find that derivative’s instantaneous rate of change.
A second derivative is obtained from the first derivative using the same concepts as the first manipulation.
For a function f(x), the first derivative can be labeled f’ (x) and the second derivative as f” (x). Essentially, the second derivative measures the rate of change of the original function.
Integrating this second derivative will result in the first derivative. Essentially, integration is the reverse of differentiation.
How do you find the original and second derivatives?
Given a function f(x), the first derivative is f’ (x), and the second derivative f” (x). The first manipulation of the original function produces the rate of change to that function f(x).
Subsequent manipulation of the first derivative produces the rate of change of the derivative.
A function is expressed as a relationship between variables and constants. For instance, f(x) = 4x^2 + 6 relates a constant 6 added to 4 times the square value of the variable x.
The first derivative will find the rate of change of x compared to f(x), and the second derivative will serve the same function as the first derivative, f’ (x).
The same rules for differentiation apply in calculus for the first, second, and subsequent derivatives. These rules are:
- The derivative of a constant is zero. In the example f(x) = 4x + 2, the differentiation of 2 will result in 0. For our previous equation f(x) = 4x^2 + 6, the differentiation of 6 also results to 0.
- The power rule of differentiation applies. It states that if f(x) = x^n (x raised to the power n), then f'(x) = nx^(n – 1). The variable constant is multiplied by the power of the variable, and the variable’s power lessens by a factor of one. In the example f(x) = 4x^2 + 6, the first derivative f’(x)= (2*4) x = 8x. To find the second derivative f” (x) and following these rules, f” (x) = 8.
- Trigonometry differentiation. The derivatives of sin x and cos x are cos x and -sin x, respectively.
The basic differentiation steps are:
- Identify the proper differentiation of a function f(x)
- Applying the rules, find the first derivative f’ (x)
- Identify the proper differentiation of the first derivative
- Differentiate the function to find the second derivative f” (x)
Here are a few examples to help you grasp the concept of first and second derivatives:
- For second power variables: If the original function is f(x) = 2x^2 + 4x + 10, the first derivative is thus f'(x) = 4x + 4 = 4(x + 1) and the second derivative f’’(x) = 4.
- For third powered variables: If the original function is f(x) = 5x^3 +2x^2 + 6x + 10, the first derivative is thus f'(x) = 15x^2 + 4x + 6 and the second derivative f’’(x) = 30x + 4.
- For fourth powered variables: If the original function is f(x) = 7x^4 + 5x^3 – 2x^2 + 6x + 10, the first derivative is thus f'(x) = 28x^3 +15x^2 – 4x + 6 and the second derivative f’’(x) = 84x^2 +30x – 4.
- A practical example of a derivative is to find the velocity and acceleration of vehicles. For instance, if the distance covered by a vehicle is provided by the function f(t) = 3t^2 + 5, the first derivative f’ (t) = 6t will provide the speed at any given time since the rate of change of distance is the speed. The second derivative will solve for acceleration which is the rate of change in speed. Acceleration f’’(t) = 6.
This YouTube video provides a neat tutorial for finding the second derivative.
How do you integrate the second derivative?
An integral is the reverse of differentiation. So, if you understood the previous concept, you’ll easily grasp the integration of the second derivative. Essentially, ∫f’’(x) = f’(x).
The basic rules for integration are (note C is a constant):
- For a constant. ∫a dx = ax + C
- For a variable. ∫x^n dx = x^(n+1)/(n+1) + C
If we apply these rules to our previous second derivatives
- For f’’(x) = 4, the integral ∫f’’(x) = 4x + C
- For f’’(x) = 30x + 4 the integral ∫f’’(x) = 30/(1+1) x^(1+1) + C = 15x^2 + C
This YouTube video provides a concise explanation of finding the integral of the second derivative.
Final thoughts
Integration and differentiation are core concepts of calculus. You can find derivatives of original functions, and further manipulation results in subsequent derivatives.
For instance, the second derivative is obtained from differentiating the first derivative. Performing an integral on the second derivative will return to the first derivative. Integrate the first derivative to obtain the original function.
