Understanding derivatives is crucial in calculus, as they represent the slope of a tangent line at any given point on a function. The derivative can be calculated for any value of x, allowing us to determine the slope of the tangent line at that specific point. To find the derivative of a function, we utilize the limit definition of a derivative, which is expressed as:
$$ f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} $$
In this formula, h represents a small change in x, and as it approaches zero, we can find the slope of the tangent line. For example, to find the derivative of the function f(x) = x^2, we substitute into the derivative formula:
$$ f'(x) = \lim_{h \to 0} \frac{(x + h)^2 - x^2}{h} $$
Expanding the numerator gives us:
$$ (x^2 + 2xh + h^2 - x^2) = 2xh + h^2 $$
Thus, the expression simplifies to:
$$ f'(x) = \lim_{h \to 0} \frac{h(2x + h)}{h} $$
By canceling h from the numerator and denominator, we arrive at:
$$ f'(x) = \lim_{h \to 0} (2x + h) $$
As h approaches zero, this simplifies to:
$$ f'(x) = 2x $$
This result indicates that the derivative of f(x) = x^2 is f'(x) = 2x, which provides the slope of the tangent line at any x value. For specific values, we can substitute directly into the derivative. For instance, at x = 1:
$$ f'(1) = 2(1) = 2 $$
And at x = -2:
$$ f'(-2) = 2(-2) = -4 $$
These calculations show that the slope of the tangent line at x = 1 is 2, while at x = -2, it is -4. The general derivative formula allows us to find the slope at any point, making it a powerful tool in calculus. Whether evaluating at x = 100 or x = -56, we can simply plug in the desired value into the derivative equation to obtain the slope of the tangent line at that point.
