Question

In Exercises \(1-10,\) find \(d y / d x\) . Use your grapher to support your analysis if you are unsure of your answer. $$y=1+x-\cos x$$

Short answer

The derivative \(dy/dx\) of the function \(y = 1 + x - \cos x\) is \[ dy/dx = 1 - \sin x \]

Step-by-step solution

Identify Function Type

The provided function \(y = 1 + x - \cos x\) is a combination of a linear function, a constant and a trigonometric function (cosine). Each component will be differentiated separately.

Differentiate Constant

The derivative of the constant \(1\) is \(0\) because the slope of a constant function is zero. Therefore, \(d/dx(1) = 0\).

Differentiate Linear Function

The derivative of linear function \(x\) is \(1\), since the slope of the line represented by the function is equal to its coefficient, which is one. Therefore, \(d/dx(x) = 1\).

Differentiate Cosine Function

The derivative of the cosine function is negative sine. Therefore, \(d/dx(\cos(x)) = -\sin(x)\).

Combine Derivatives

Combine the computed derivatives to get the derivative for the entire function. Therefore, the derivative for \(y = 1+x-\cos x\) is \(0 + 1 - \sin(x)\).