Let f(x)=| cos x sin x cos x cos 2 x sin 2 x 2 cos 2 x cos 3 x s

step 1 In this question, we are given with the determinant that is having cos x, sine x, cos x, cos 2x,

Let f(x)=| cos x sin x cos x cos 2 x sin...

Let $\quad f(x)=\left|\begin{array}{lll}\cos x & \sin x & \cos x \\ \cos 2 x & \sin 2 x & 2 \cos 2 x \\ \cos 3 x & \sin 3 x & 2 \cos 3 x\end{array}\right|$ Then find the value of $f^{\prime}\left(\frac{\pi}{2}\right)$.

Key Concepts

Trigonometric Functions and Their Properties

Trigonometric functions, like sine and cosine, are fundamental in mathematics due to their periodic behavior and their relationships within right triangles. Their properties, including periodicity, symmetry, and specific values at key angles, are crucial when simplifying expressions and calculating derivatives involving these functions.

Evaluation of Functions at Specific Points

Evaluating a function at a specific point involves substituting the point into the function and simplifying the resulting expression using known values, identities, or other properties of the function's components. This technique is widely used to find exact values of functions, especially when those functions incorporate trigonometric expressions or determinants.

Determinant of a Matrix

The determinant is a scalar value associated with a square matrix that conveys important information about the matrix, including whether it is invertible and the scaling factor it applies to volume in linear transformations. It is computed using various methods, such as expansion by minors or row-reduction, and plays a critical role in many areas of algebra and calculus.

Differentiation of Functions Defined by Determinants

When a function is defined as the determinant of a matrix with entries that are functions of a variable, its derivative can be determined by differentiating each entry of the matrix and applying the rules of differentiation to the determinant expression. This process often involves techniques such as differentiating under the determinant sign and simplifying the resulting expression.

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