DERIVATIVE f(x)=4t+1 (x,x+2)

To find the derivative of the function f(x) = 4t + 1 with respect to x, we will use the formula for finding the derivative of a linear function. In this case, we are given that x and t are related by x = t + 2.

First, we need to express the function f(x) in terms of t. Since x = t + 2, we can solve for t by subtracting 2 from both sides of the equation: t = x - 2.

Now we can substitute t = x - 2 into the function f(x): f(x) = 4(x - 2) + 1.

Expanding the function, we have: f(x) = 4x - 8 + 1 = 4x - 7.

To find the derivative df/dx, we differentiate f(x) with respect to x by applying the power rule. The power rule states that if we have a term of the form ax^n, its derivative with respect to x is given by nx^(n-1).

In this case, the derivative of 4x is 4, and the derivative of -7 (a constant) is 0. Therefore, the derivative of f(x) = 4x - 7 with respect to x is simply 4.

So, the derivative of f(x) = 4t + 1 with respect to x is df/dx = 4.