Basic Derivative
The basic derivative is a fundamental concept in calculus that involves finding the instantaneous rate of change of a function at any given point
The basic derivative is a fundamental concept in calculus that involves finding the instantaneous rate of change of a function at any given point. It represents the slope or steepness of the function at that point.
To find the derivative of a function, you can use the derivative rules, such as the power rule, constant rule, sum rule, and product rule.
Let’s start with the power rule, which is used to find the derivative of functions in the form of f(x) = x^n, where n is a constant.
Suppose we have the function f(x) = x^2. The power rule states that the derivative of x^n is equal to n times x^(n-1), where n is a constant.
Applying this to our function, we have:
f'(x) = 2x^(2-1) = 2x
Therefore, the derivative of f(x) = x^2 is f'(x) = 2x.
Now, let’s consider the constant rule, which states that if a function is multiplied by a constant, the derivative of the constant times the function is equal to the constant times the derivative of the function.
For example, let’s find the derivative of g(x) = 5x^3.
Using the constant rule, we can write the derivative as:
g'(x) = 5 * (d/dx) (x^3)
Now, we can apply the power rule to find the derivative of x^3:
g'(x) = 5 * 3x^(3-1) = 5 * 3x^2 = 15x^2
Therefore, the derivative of g(x) = 5x^3 is g'(x) = 15x^2.
Next, let’s look at the sum rule, which allows us to find the derivative of the sum of two functions.
Suppose we have h(x) = f(x) + g(x), where f(x) = 4x^2 and g(x) = 3x.
To find the derivative of h(x), we can find the derivative of each function separately and then add them together:
h'(x) = (d/dx) (4x^2) + (d/dx) (3x)
Using the power rule for f(x), we have:
(d/dx) (4x^2) = 4 * 2x^(2-1) = 8x
Applying the power rule to g(x), we have:
(d/dx) (3x) = 3 * 1x^(1-1) = 3
Therefore, the derivative of h(x) is:
h'(x) = 8x + 3
Lastly, let’s discuss the product rule, which is used to find the derivative of the product of two functions.
Consider the function p(x) = f(x) * g(x), where f(x) = 2x^3 and g(x) = 3x.
To find the derivative of p(x), we can use the product rule, which states that the derivative of the product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.
The derivative of p(x) is given by:
p'(x) = f'(x) * g(x) + f(x) * g'(x)
Using the power rule for f'(x) and g'(x), we have:
f'(x) = 3 * 2x^(3-1) = 6x^2
g'(x) = 1 * 3x^(1-1) = 3
Substituting these values into the product rule equation, we get:
p'(x) = (6x^2) * (3x) + (2x^3) * (3)
Simplifying further, we have:
p'(x) = 18x^3 + 6x^3 = 24x^3
Therefore, the derivative of p(x) = 2x^3 * 3x is p'(x) = 24x^3.
These are the basic derivative rules that can be used to find the derivative of various functions. It is important to practice these rules and understand their applications in order to solve more complex derivative problems in calculus.
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