Derivatives using the limit definition of the dervative

Compute $f'(x)$ using the limit definition of the derivative $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ for the following 1. f(x) = 3 2. f(x) = 3x-1 3. f(x) = $x^2 + x$ 4. f(x) = $\sqrt(x)$ 5. f(x) = 1/x

When computing derivatives using the limit definition, we are essentially looking at how a function changes as its input changes by a very small amount. The idea is to measure the rate of change of the function at any given point by examining the difference between the function's value at that point and a nearby point. As the gap between these two points shrinks to zero, we get the derivative, which tells us the slope or steepness of the function at that exact point.

In general, the approach starts by finding the difference between the function's value at two close points. We then divide this difference by how much the input changed, which gives us an average rate of change over that small interval. Finally, we take the limit of this expression as the interval becomes infinitesimally small. This limit process is key to finding the instantaneous rate of change, which is the derivative.

Different types of functions behave differently when you apply this method. For example, constant functions don’t change at all, so their derivative is zero. Linear functions, like straight lines, have the same slope everywhere, so their derivative is constant. For more complex functions, like quadratic or square root functions, the derivative changes depending on the input, reflecting how the function curves or levels out at different points. The limit-based approach works the same in all cases, but the algebraic steps can vary depending on the complexity of the function.

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