Question

Find the derivatives of the given functions and graph along with the function to ensure your answer is correct.\([T] \sinh ^{6}(x)\)

Short answer

The derivative is \( 6\sinh^5(x)\cosh(x) \).

Step-by-step solution

Understand the Function

The given function is \( y = \sinh^6(x) \). We need to find the derivative \( \frac{dy}{dx} \).

Apply the Chain Rule

To differentiate \( y = \sinh^6(x) \), recognize it as \( (\sinh(x))^6 \). According to the chain rule, if you have \( u(x)^n \), then \( \frac{d}{dx}[u(x)^n] = n\cdot u(x)^{n-1} \cdot \frac{du}{dx} \). Here, \( u(x) = \sinh(x) \) and \( n = 6 \).

Differentiate the Inner Function

We first find the derivative of the inner function \( u = \sinh(x) \), which is \( \frac{du}{dx} = \cosh(x) \).

Differentiate the Outer Function

Using the chain rule, differentiate the outer function. Our expression becomes:\[ \frac{d}{dx}[\sinh(x)^6] = 6 \cdot \sinh(x)^5 \cdot \cosh(x) \].

Verify the Derivative with Graphs

Now plot both the original function \( y = \sinh^6(x) \) and its derivative \( y' = 6 \cdot \sinh(x)^5 \cdot \cosh(x) \) using a graphing tool. Verify that the derivative graph represents the slope of the tangent lines to the original function at any point \( x \).

Key concepts

Chain Rule

The Chain Rule is a fundamental tool in calculus used for finding the derivative of a composite function. When dealing with functions nested within other functions, understanding how to use the Chain Rule is essential. It allows us to differentiate complex expressions.
The Chain Rule states that if you have a composite function like \( y = f(g(x)) \), then the derivative \( \frac{dy}{dx} \) is given by \( f'(g(x)) \cdot g'(x) \).

• First, identify the inner function \( u(x) \) and the outer function \( f(u) \).
• Find the derivative of both the inner and the outer functions.
• Multiply these derivatives as described by the chain rule.

In our example, the function \( y = \sinh^6(x) \) can be rewritten as \( [u(x)]^6 \), where \( u(x) = \sinh(x) \). The Chain Rule becomes crucial as it helps in breaking down the differentiation process into manageable steps.
For the derivative, we first differentiate \( u(x) \) to get \( \cosh(x) \), then apply the Chain Rule formula to the outer function. This yields \( 6 \cdot \sinh(x)^5 \cdot \cosh(x) \), seamlessly applying the rule to obtain the result.

Hyperbolic Functions

Hyperbolic functions, like exponential and trigonometric functions, have a unique set of properties making them particularly useful in certain areas of calculus and hyperbolic geometry. The primary hyperbolic functions are \( \sinh(x) \), \( \cosh(x) \), \( \tanh(x) \), and their inverses.
The function \( \sinh(x) \), or hyperbolic sine, is defined as \( \sinh(x) = \frac{e^x - e^{-x}}{2} \). This function grows exponentially but coordinates with the unit hyperbola, unlike the unit circle in trigonometry.

• Trigonometric Similarity: These functions relate closely to trigonometric ones but with hyperbolic identities, like \( \cosh^2(x) - \sinh^2(x) = 1 \).
• Derivative: When differentiating \( \sinh(x) \), its derivative is \( \cosh(x) \), which is like how the derivative of \( \sin(x) \) is \( \cos(x) \).

These identities are critical, particularly in calculus problems involving the Chain Rule and differentiation, assisting in methodically solving for derivatives.

Graphical Verification

Graphical verification is a helpful practice to confirm calculus solutions visually. After deriving the function \( y = \sinh^6(x) \), plotting both the original and derivative functions on a graph can offer intuitive insight.
The process involves graphing \( y = \sinh^6(x) \) and its derivative \( y' = 6 \cdot \sinh(x)^5 \cdot \cosh(x) \). Observing these graphs reveals several things:

• Slope Confirmation: The derivative plot represents the gradient or slope of the tangent line to the original function at every point \( x \).
• Behavior Matching: Notice how peaks and troughs on the \( y \) graph correspond to zero crossings on the \( y' \) graph. This aligns with where tangents are horizontal.

Such verification helps ensure that the derivative function accurately captures the rate of change, offering a practical check against the algebraic solution.