Summary of Differential Calculus
Differential calculus is the study of slope, the tangent, and the normal of the curve and rate of change on the curve by means of derivatives and differentials. The derivative can be shown with , , and . Note that is a whole rather than a friction. The process of finding the derivatives is called differentiation. The method contains in finding the derivative is the limit method which can also be called the first principles. As we know, the slope of should include the coordinates of two points and and the formula for the slope is . If we choose a point on the curve, in order to, find the slope, we have to choose another point. Therefore, the slope of the curve is . Then, make the h approaches 0 and the slope will be that of the point on the curve. Substitute the value of x into the slope, then, the slope of the certain point can be figured out. Also, if the question is to find out the slope of a certain point on the curve of the function f(x), another method can be utilized. , as x approaches a, the derivative will be calculated. During each of the calculation, the denominator of the function should be eliminated, so the numerator of the equation should be transformed. When simplify the formula, some practical rules were found which make the simplification much easier. If the curve is a power function, the formula should just be expanded and eliminated. If the curve is a rational function, the denominator in the numerator should be eliminated and then the formula will be simplified. If the function is in the surd form, them, the square root and other root should be eliminated. After several calculation of the derivative of the power function, a rule is figured out that the derivative of the function is . A special kind of the function is the constant function, whose derivative is 0. For all the functions, which can be transformed in this format, this rule can be the approach to find their derivative. If the n is a polynomial...
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