Problem 42
Question
In Problems \(1-58\), find the derivative with respect to the independent variable. $$ f(x)=\frac{1}{\sin ^{2} x+\cos ^{2} x} $$
Step-by-Step Solution
Verified Answer
The derivative is 0.
1Step 1: Identify the Trigonometric Identity
The expression \( \sin^2 x + \cos^2 x \) is a well-known trigonometric identity that simplifies to 1. Thus, \( f(x) = \frac{1}{1} = 1 \).
2Step 2: Differentiate the Constant Function
Since \( f(x) = 1 \) is a constant function, its derivative with respect to \( x \) is \( 0 \). Therefore, \( f'(x) = 0 \).
Key Concepts
Trigonometric identitiesConstant functionDerivative calculation
Trigonometric identities
Trigonometric identities are fundamental tools in calculus and trigonometry. They are equations involving trigonometric functions that are true for most angles. One of the most well-known identities is the Pythagorean identity:
When you encounter expressions like \( \sin^2 x + \cos^2 x \), recognizing it as the Pythagorean identity allows you to simplify problems immediately. For the given exercise, applying this identity allows \( f(x) = \frac{1}{\sin^2 x + \cos^2 x} \) to reduce to \( f(x) = \frac{1}{1} = 1 \). This simplification can drastically change the complexity of a problem, often reducing it to easier functions to work with.
- \( \sin^2 x + \cos^2 x = 1 \)
When you encounter expressions like \( \sin^2 x + \cos^2 x \), recognizing it as the Pythagorean identity allows you to simplify problems immediately. For the given exercise, applying this identity allows \( f(x) = \frac{1}{\sin^2 x + \cos^2 x} \) to reduce to \( f(x) = \frac{1}{1} = 1 \). This simplification can drastically change the complexity of a problem, often reducing it to easier functions to work with.
Constant function
A constant function is one where the output value remains the same regardless of the input. It does not change with respect to the variable. Mathematically, if \( f(x) = c \), where \( c \) is a constant, the function's graph is a horizontal line at \( y = c \).
In the exercise above, after applying the trigonometric identity, the function became \( f(x) = 1 \). This is a classic example of a constant function because no matter the value of \( x \), \( f(x) \) is always 1.
Understanding constant functions is crucial in calculus as they represent scenarios where changes in variables have no impact on the function's output. It makes differentiation straightforward.
In the exercise above, after applying the trigonometric identity, the function became \( f(x) = 1 \). This is a classic example of a constant function because no matter the value of \( x \), \( f(x) \) is always 1.
Understanding constant functions is crucial in calculus as they represent scenarios where changes in variables have no impact on the function's output. It makes differentiation straightforward.
Derivative calculation
Calculating derivatives is a central part of calculus, as it measures how a function changes with respect to its variables. The derivative of a function \( f(x) \) is often denoted as \( f'(x) \) or \( \frac{df}{dx} \).
For constant functions, like the one derived in the exercise, the derivative is always zero. This is because with a constant function, there is no change in the output as the input changes.
Such results are immediately useful: when you recognize a function as constant, you know the derivative without performing complex calculations. This understanding can simplify more complicated problems in calculus.
For constant functions, like the one derived in the exercise, the derivative is always zero. This is because with a constant function, there is no change in the output as the input changes.
- If \( f(x) = c \), then \( f'(x) = 0 \).
Such results are immediately useful: when you recognize a function as constant, you know the derivative without performing complex calculations. This understanding can simplify more complicated problems in calculus.
Other exercises in this chapter
Problem 42
4Suppose a particle moves along a straight line. The position at time \(t\) is given by $$ s(t)=3 t-t^{2}, \quad t \geq 0 $$ where \(t\) is measured in seconds
View solution Problem 42
Find \(\frac{d y}{d x}\) by applying the chain rule repeatedly. \(y=\left(1+2(x+3)^{4}\right)^{2}\)
View solution Problem 42
Differentiate the functions in Problems 1-52 with respect to the independent variable. $$ h(t)=4^{2^{3}-t} $$
View solution Problem 42
Assume that \(f(x)\) and \(g(x)\) are differentiable at \(x\). Find an expression for the derivative of \(y .\) $$ y=[f(x)-3] g(x) $$
View solution