Derivative of Cosine of x Squared

Find the derivative of $\cos(x^2)$.

Differentiating a function such as cosine of x squared involves applying the chain rule, a fundamental technique in calculus for finding the derivative of composite functions. The chain rule allows us to differentiate functions embedded within other functions by systematically breaking them down and analyzing their individual components. In this particular problem, recognizing that cosine of x squared has an inner function (x squared) and an outer function (cosine) will be crucial for correctly applying the chain rule.

To apply the chain rule, we first take the derivative of the outer function, cosine, which gives us negative sine, keeping the inside function, x squared, unchanged temporarily. Next, we differentiate the inner function, x squared, resulting in a linear function of x. By multiplying these results together, we arrive at the derivative of the entire composite function. Successfully navigating through these steps requires an understanding of both the general process of differentiation and the specific mechanics of the chain rule.

As you gain proficiency with the chain rule, it serves as a versatile and powerful tool for handling more complex derivatives encountered in calculus, whether in theoretical aspects or practical applications. This particular problem exemplifies a basic yet essential exercise in applying these differentiation principles, reinforcing the concept that many challenging calculus problems can be approached by systematically breaking them down into simpler components.

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