Find the exact value of the derivative, $f'(a)$, and the slope of the tangent line for any function at a given point $x=a$ with a full step-by-step solution.
Finding the derivative $f'(x)$ and evaluating at $x=a$...
In calculus, the derivative of a function $f(x)$ at a specific point $x=a$, denoted $f'(a)$, has a vital geometric meaning: **it represents the instantaneous slope of the function at that exact point.**
The value of $f'(a)$ is equivalent to the slope ($m$) of the line tangent to the curve $f(x)$ at the point $(a, f(a))$. This concept is the foundation of differential calculus. [Image of the tangent line at a point on a curve]
The calculation is a two-step process:
For example, if $f(x) = x^2$, the general derivative is $f'(x) = 2x$. If we want the derivative at $x=3$, we plug 3 into $f'(x)$: $f'(3) = 2(3) = 6$. The slope of the tangent line at $x=3$ is 6.
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