Derivative of tan(sin x) - cos(cos x) with Respect to x: A Step-by-Step Approach

Introduction to Trigonometric Functions and Derivatives

When working with calculus, one often encounters complex functions involving trigonometric functions such as sine, cosine, and tangent. To solve problems involving these functions, it is crucial to understand and apply the rules of differentiation, particularly the chain rule. This article will walk you through finding the derivative of the composite function (tan(sin x) - cos(cos x)) with respect to (x).

The Chain Rule: A Fundamental Tool in Calculus

The chain rule is a key concept in calculus used to find the derivative of a composition of functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function, multiplied by the derivative of the inner function. Mathematically, it is expressed as:

dfdx ddu fu ? ddx u

Where (u g(x)) is the inner function and (f(u)) is the outer function.

Derivative of tan(sin x)

The first part of the given function is (tan(sin x)). We apply the chain rule as follows:

dfdx ddu tanu ? ddx sinx ( secu2 ) ? ( cosx ) secsin2x ? cosx

Derivative of cos(cos x)

The second part of the given function is (-cos(cos x)). Again, we apply the chain rule:

dfdx ddu cosu ? ddx cosx ( - sinu ) ? ( - sinx ) - sincosx ? sinx

Combining the Results

To find the derivative of the entire function (tan(sin x) - cos(cos x)), we combine the derivatives of the two parts:

d tan sinx - cos cosx dx secsin2x ? cosx - ( sincosx ? sinx ) secsin2x ? cosx - sincosx ? sinx

The final derivative is:

d tan sinx - cos cosx dx secsin2x ? cosx - sincosx ? sinx - sincosx ? sinx - secsin2x ? cosx

Conclusion

Understanding and applying the chain rule is crucial for solving complex derivative problems involving trigonometric functions. By breaking down each part of the function and applying the chain rule step-by-step, we can find the derivative of the given function (tan(sin x) - cos(cos x)).