Derivative of a Linear Function
To find the derivative of a linear function, we first need to understand what a linear function is
To find the derivative of a linear function, we first need to understand what a linear function is. A linear function is a function in the form f(x) = mx + b, where m is the slope of the line and b is the y-intercept.
Now, let’s find the derivative of a linear function f(x) = mx + b. The derivative of a function gives us the rate at which the function is changing at any given point.
To find the derivative of f(x), we need to use the power rule of differentiation. The power rule states that the derivative of x^n is nx^(n-1), where n is any real number.
In the case of f(x) = mx + b, we can see that x has a power of 1, since it is to the power of 1 implicitly, and m is a constant. Using the power rule, we can differentiate each term separately.
The derivative of mx is m * (x^(1-1)) = m * x^0 = m.
The derivative of b is 0, since b is a constant and its derivative is always zero.
Therefore, the derivative of f(x) = mx + b is f'(x) = m.
What does this mean? It means that the slope, m, of the original linear function f(x), is equal to the derivative of the function. This makes intuitive sense since the slope of a linear function is constant and does not vary with x.
So, in summary, the derivative of a linear function f(x) = mx + b is equal to the slope of the line, m.
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