Question

Find the derivative. $$D_{x}(7-5 x)$$

Short answer

-5

Step-by-step solution

Apply the derivative operator to each term

The derivative of a constant term is 0, and the derivative of a term with x is the coefficient multiplied by the power of x reduced by 1. Since the term '-5x' is linear, its derivative will just be its coefficient.

Differentiate each term individually

Differentiate the constant 7 to get 0 and differentiate '-5x' to get -5 as the power of x is 1, and reducing it by 1 leaves us with x to the power of 0, which is 1.

Combine the derivatives of all terms

Combine the derivatives of each term to find the derivative of the entire function. The derivative of 7 is 0, and the derivative of -5x is -5, so the combined result is simply the derivative of -5x.

Key concepts

Differentiation Rules

Differentiation is a fundamental concept in calculus that deals with the analysis of functions to find the rate at which they change. When finding the derivative of a function, certain rules can be systematically applied to make the process straightforward. The most basic rule is that the derivative of a constant, a number without any variable like the number 7 in our exercise, is always zero. This happens because a constant does not change, and hence, its rate of change is zero.

Another important differentiation rule is the linearity of the derivative, which allows us to take the derivative of each term separately when they are added or subtracted. This means in a function like the one in our exercise, we can differentiate 7 and \( -5x \) independently and then add their derivatives together for the final answer. By adhering to these basic rules every time, you ensure consistent and correct results when differentiating functions.

Power Rule

The power rule is a quick way to find the derivative of a function that involves a variable raised to a power. The rule states that if you have a term in the form \( x^n \) where \( n \) is any real number, its derivative is \( nx^{n-1} \). Therefore, to differentiate a term like \( -5x \) (which is implied to be \( -5x^1 \) ), we identify the exponent \( n \) as 1. Following the power rule, we multiply the coefficient by the current exponent to maintain the \( -5 \) and then decrement the exponent by one. Since any number to the power of 0 is 1, we end with \( -5x^0 \) or just \( -5 \). This rule enables us to handle more complex functions with ease, as calculating derivatives becomes much more manageable, especially for polynomial functions where each term is a power of \( x \) with a coefficient.

Linear Function Derivative

A linear function is a function of the form \( f(x) = ax + b \), where \( a \) and \( b \) are constants. The derivative of a linear function is special because it represents a constant rate of change. In the case of our exercise, \( D_{x}(7-5x) \) is a simple representation of a linear function where \( a = -5 \) and \( b = 7 \). As per the power rule, the \( x \) term here is differentiable, and its derivative is the constant \( a \) itself, so the derivative of \( -5x \) is \( -5 \).

Since the derivative of a constant like \( b \) is zero, we ignore it in the differentiation process. The takeaway is quite powerful for linear functions: regardless of the value of \( x \) , the derivative will always be the coefficient \( a \) of the \( x \) term. This not only makes calculating derivatives of linear functions straightforward but also signifies that the slope of the line represented by a linear function is constant.