Question

Find the derivatives of the given functions and graph along with the function to ensure your answer is correct.\([\mathrm{T}] \cosh ^{2}(x)+\sinh ^{2}(x)\)

Short answer

The derivative is \(f'(x) = 2\sinh(2x)\). Graphs will confirm this visual observation.

Step-by-step solution

Simplify the Function Using Hyperbolic Identity

Recall that there's a fundamental hyperbolic identity:\[\cosh^2(x) - \sinh^2(x) = 1\]However, the function given is:\[\cosh^2(x) + \sinh^2(x)\]In this case, it simplifies to:\[(\cosh^2(x) - \sinh^2(x)) + 2\sinh^2(x) = 1 + 2\sinh^2(x)\]

Differentiate the Simplified Function

The function is now: \[f(x) = 1 + 2\sinh^2(x)\]To find the derivative, recall that:\[\frac{d}{dx} \sinh(x) = \cosh(x)\]And apply the chain rule:\[\frac{d}{dx}(\sinh^2(x)) = 2\sinh(x) \cdot \frac{d}{dx}(\sinh(x)) = 2\sinh(x) \cdot \cosh(x)\]Thus, the derivative of the function is:\[f'(x) = 2 \cdot 2\sinh(x) \cosh(x) = 4\sinh(x)\cosh(x)\]

Verify the Derivative Using the Double Angle Identity

Using the hyperbolic double angle identity:\[2\sinh(x)\cosh(x) = \sinh(2x)\]The derivative can be rewritten as:\[f'(x) = 2\sinh(2x)\]This confirms the result using a known identity.

Graph the Function and Its Derivative

Graph the original function, \(f(x) = 1 + 2\sinh^2(x)\), and its derivative, \(f'(x) = 4\sinh(x)\cosh(x)\) or equivalently \(2\sinh(2x)\).The graph of \(f(x)\) will be quadratic, reflecting the positive contribution of \(\sinh^2(x)\), while \(f'(x)\) will show hyperbolic sine characteristics.

Key concepts

Hyperbolic Functions

Hyperbolic functions are analogs of the well-known trigonometric functions but are based on hyperbolas rather than circles. They include functions like hyperbolic sine (\[ \sinh(x) = \frac{e^x - e^{-x}}{2}\])and hyperbolic cosine (\[ \cosh(x) = \frac{e^x + e^{-x}}{2}\]). These expressions reveal how hyperbolic functions relate to exponential functions, and they are often used in calculus due to their unique identities and derivatives.**Key Hyperbolic Identity**One important identity to remember is:\[\cosh^2(x) - \sinh^2(x) = 1\]This identity is similar to the Pythagorean identity for sine and cosine in circular trigonometry, and it plays a significant role in simplifying expressions and solving differentiation problems involving hyperbolic functions.

Chain Rule

The chain rule is a powerful tool in calculus used to differentiate composite functions. Whenever you see a function nested inside another function, the chain rule is your go-to method.**Applying to Hyperbolic Functions**For instance, if you need to differentiate \[\sinh^2(x)\]you treat it as a composite function. The outer function is the square (\[(... )^2\]), and the inner function is \[\sinh(x)\]. Using the chain rule means differentiating the outer function while maintaining the inner function and then multiplying by the derivative of the inner function.**Example: Differentiating \[\sinh^2(x)\]**1. Differentiate the outer function: \[2\sinh(x)\](noting that the exponent becomes a coefficient).2. Multiply by the derivative of the inner function, \[\cosh(x)\].3. Result is \[2\sinh(x)\cosh(x)\].This methodology not only helps in differentiating complex hyperbolic expressions but also ensures precision in calculus.

Graphing Functions

Graphing functions and their derivatives is an essential skill in calculus. It visually confirms the behavior of functions and the accuracy of derived expressions.**Graphing Hyperbolic Functions**When you graph the original function \[f(x) = 1 + 2\sinh^2(x)\], you'll notice it behaves similarly to a quadratic function. This is due to the \[\sinh^2(x)\]term, which has a positive contribution, causing the graph to curve upward.**Graphing the Derivative**The derivative \[f'(x) = 4\sinh(x)\cosh(x)\]resembles the hyperbolic sine function. The graph of the derivative confirms its hyperbolic sine feature through exponential growth. This can alternatively be expressed as \[2\sinh(2x)\], showcasing hyperbolic characteristics.Visualizing these functions on a graphing calculator or software deepens understanding and illustrates function behavior.

Derivative Verification

Ensuring that a calculated derivative is indeed correct is crucial in mathematics.**Using Identities for Verification**To verify the derivative \[f'(x) = 4\sinh(x)\cosh(x)\], we use the hyperbolic double angle identity. The identity states:\[2\sinh(x)\cosh(x) = \sinh(2x)\]When applied, this simplifies the derivative to \[2\sinh(2x)\]. This confirms our result matches known identities, reinforcing the accuracy.Verification is not only about checking mathematical correctness. It's also a method to affirm our understanding of both the techniques and the underlying principles involved in solving calculus problems.