Differentiation I: An Overview of Differential Calculus
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Differential calculus is the study of the slopes of functions and their application to real world problems. Recall that the slope of a straight line is its "rise" over its "run":
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More importantly, we can determine the slope anywhere along the straight line and always get the same answer.However, for any other function, the slope can be different depending upon where we look. For example, the picture below shows that the function <Sign in to see all the formulas> has a slope of 10 at $x=5$, 0 at $x=0$, and 4 at $x=2$. So depending on where we are at on the xaxis, our slope will be different from point to point.
There are typically two common notations for the slope of a function. The slope of a function at a point $x$ of a function $f(x)$ can be denoted by
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and is typically called the "derivative" of the function or more formally " the derivative of 'f' with respect to 'x'."
So why is the slope of a function so important that we have an entire branch of mathematics for it? Because there are many real world phenomenon that are described by the slope of a function. Many things in life depend upon something else. In those situations we are also wanting to know how one variable changes with respect to another. Here are some examples:
Derivatives In The Real World 








So now that we have a basic understanding of derivatives and what they are used for, here is how we actually find the derivative of a function.