Given $y=x+1+x+1$, find $\frac{dy}{dx}$.

The derivative of the equation y = x + 1 + x + 1 simplifies to 2 x + 2 , leading to the result d x d y ​ = 2 . This result indicates that the rate of change of y with respect to x is constant. Therefore, for any change in x , y will increase by 2 times that change.
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Answered by Anonymous | 2025-07-04

Simplify the given equation: y = x + 1 + x + 1 = 2 x + 2 .
Apply the power rule to differentiate y with respect to x : d x d y ​ = d x d ​ ( 2 x + 2 ) .
Calculate the derivative of 2 x as 2 and the derivative of 2 as 0.
Obtain the final derivative: d x d y ​ = 2 ​ .

Explanation

Problem Analysis We are given the equation y = x + 1 + x + 1 and asked to find its derivative with respect to x , which is denoted as d x d y ​ .

Simplifying the Equation First, simplify the equation:


y = x + 1 + x + 1 = 2 x + 2

Applying Differentiation Now, differentiate y with respect to x . Recall the power rule: d x d ​ ( a x n ) = na x n − 1 , where a is a constant and n is the exponent. Also, the derivative of a constant is 0.

So, d x d y ​ = d x d ​ ( 2 x + 2 ) = d x d ​ ( 2 x ) + d x d ​ ( 2 )

Calculating the Derivative Applying the power rule, we have:

d x d ​ ( 2 x ) = 2 ⋅ d x d ​ ( x ) = 2 ⋅ 1 ⋅ x 1 − 1 = 2 ⋅ 1 ⋅ x 0 = 2 ⋅ 1 ⋅ 1 = 2
And, the derivative of the constant 2 is 0:
d x d ​ ( 2 ) = 0
Therefore, d x d y ​ = 2 + 0 = 2

Final Answer Thus, the derivative of y with respect to x is 2.

Examples
In physics, if y represents the position of an object at time x , then d x d y ​ represents the object's velocity. If y = 2 x + 2 , then the velocity is constant and equal to 2, meaning the object is moving at a steady pace. This concept is fundamental in understanding motion and rates of change in various real-world scenarios.

Answered by GinnyAnswer | 2025-07-04