Question

In Exercises \(1-6,\) find \(d y / d x\). $$y=-x^{2}+3$$

Short answer

The derivative of the function \(y = -x^{2}+3\) in respect to \(x\) is \(dy/dx = -2x\)

Step-by-step solution

Identify the Function

Identify the given function which is \(y=-x^{2}+3\). The power of x in this equation is \(2\), and the coefficients include \(-1\) and \(3\)

Apply the Power Rule to First Term

The power rule states that the derivative of \(x^n\) is \(n*x^{n-1}\). Apply this rule to the first term: \(-x^2\). Multiply the coefficient of the x-term by the exponent: \(-1*2 = -2\). The power of x is then decreased by 1: \(x^{2-1} = x\). The derivative of \(-x^2\) is thus \(-2x\).

Apply the Power Rule to Second Term and Sum

The second term, \(3\), is a constant and its derivative is simply zero since constants disappear in differentiation. Finally, you sum up the results of the first and second term, having \(dy/dx = -2x + 0 = -2x\).