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Derivative of a Rational Expression

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Find the derivative of 1(x2+8)3\frac{1}{(x^2 + 8)^3}.

This problem requires finding the derivative of a rational expression, which is a common task in calculus. The function given here is particularly interesting because it involves a composite function that will require the use of the chain rule to differentiate effectively. The chain rule is a fundamental tool in calculus when dealing with composite functions. It allows us to differentiate compositions by taking the derivative of the outer function and multiplying it by the derivative of the inner function.

In this problem, the outer function is something raised to the third power, and the inner function is the squared sum of a variable with a constant. Differentiating these kinds of functions involves careful application of the chain rule, paying attention to the power rule for derivatives, and employing algebraic manipulation to simplify the function before and after differentiation. Understanding these concepts is crucial because mastering the technique of differentiating composite functions lays the groundwork for more complex calculus problems that you might encounter later in the subject.

Moreover, such problems are also a good exercise to test the understanding of function transformations and algebraic simplifications, which often come hand in hand with calculus problems. While this problem is representative of a typical task in single-variable calculus, it also prepares students for more advanced applications, such as in multivariable calculus, where these principles are extended to functions of several variables.

Posted by grwgreg 15 days ago

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