If f(x)=| 1+sin ^2 x cos ^2 x 4 sin 2 x sin ^2 x 1+cos ^2 x 4 sin 2 x

If f(x)=| 1+sin ^2 x cos ^2 x 4 sin 2 x sin...

Key Concepts

Limit Evaluation in Calculus

Limit evaluation is the process of determining the value that a function approaches as the input approaches a particular point. This concept is fundamental in calculus, notably in the context of derivatives and continuity, and is used to analyze the behavior of functions under transformations and differentiations.

Trigonometric Function Approximations

Trigonometric functions, such as sine and cosine, can be approximated by their Taylor series expansions for small values of the variable. These approximations help in converting non-linear trigonometric expressions into polynomial forms, thereby simplifying the process of differentiation and limit evaluation.

Differentiation of Matrix-Dependent Functions

When a function is defined as a determinant of a matrix whose entries are functions of a variable, the function inherits a composite structure that can be differentiated by applying rules of differentiation to each of the constituent functions. This concept involves applying the chain rule and understanding how small changes in the parameters affect the overall determinant.

Determinants

A determinant is a scalar value computed from a square matrix that encapsulates important properties of the matrix, such as its invertibility and how it scales volumes in linear transformations. It is central to linear algebra, often calculated using cofactors and expansion, and it has properties like linearity that are useful in simplification of expressions.

Taylor Series Expansion

Taylor series expansion is a method for approximating functions near a specific point by a power series. In calculus, this tool is crucial for simplifying complex functions into polynomials which are easier to differentiate and integrate, especially when assessing the behavior of functions near points like zero.

Transcript

00:01 So in the given question we have a determinant that is given as f of x is equal to the determinant 1 plus sine square x cos square x 4 sine 2 x this is the first row in the second row we have sine square x 1 plus cos square x and 4 sine 2 x in the third row we have 2 minus sine square x minus cos square x and 1 plus 4 sine 2 x so this is what is given as f of x right and we are told to find limit x tends to 0 limit x tends to 0 limit x tends to 0 f double dash x which is the second derivative of f of x divided by x so this is what we need to find so what we can do here first is we can simplify this determinant first right so we can do the column operations on the column operation there on this determinant and we can do the column operation c1 changes to c1 plus c2.

01:33 So if we do this the determinant would change to 2 since we have 1 plus sine square x plus cos square x so we are going to use the identity sine square x plus cos square x is equal to 1 right and if we use this identity we can substitute for sine square x plus cos square x as 1 so we have 1 plus 1 plus 1 plus 2 2.

01:57 2 then we have similarly we have 2 in the as the second element in the first column and 1 over here and then we have the rest of the determinant is the same right so let's write this 4 sine 2 x 4 sine 2 x and 1 plus 4 sine 2 x and 1 plus 4 sine 2 x now what we can do is we can take this determinant and do a row operation in order to evaluate this more easily.

02:42 So the row operation is the row operation is row 2 changes to row 2 minus row 1 and this would give us 2 cos square x 4 sine 2 x.

03:04 The second row would change to 010 and the third row is the same which is 1 minus cosquare x 1 plus 4 sine 2 x and now we can expand this right...

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