Question

Find the derivatives of the following functions. $$f(x)=\sinh 4 x$$

Short answer

Answer: The derivative of $$f(x) = \sinh 4x$$ is $$f'(x) = 4 \cosh{4x}$$.

Step-by-step solution

Identify the inner function

The given function $$f(x) = \sinh 4x$$ has the form of a composite function. The inner function, denoted by $$u(x)$$, is $$u(x) = 4x,$$ with $$u'(x) = 4.$$

Identify the outer function

The outer function is denoted by $$g(u(x))$$. In this case, the outer function is $$g(u) = \sinh u.$$

Differentiate the outer function

We need to find the derivative of the outer function with respect to $$u$$, which is denoted by $$g'(u)$$. $$g'(u) = \frac{d}{du} \sinh u = \cosh u$$

Apply the chain rule

The chain rule states that: $$f'(x) = (g'(u(x))) * (u'(x)) $$ We'll use the derivatives we found in Steps 1 and 3: $$f'(x) = (\cosh(4x)) * (4)$$

Write the final answer

The derivative of $$f(x) = \sinh{4x}$$ is: $$f'(x) = 4 \cosh{4x}$$

Key concepts

Chain Rule

Understanding the chain rule is essential for differentiating composite functions, and it comes in handy when we encounter functions within functions. Essentially, the chain rule allows us to take the derivative of an outer function and multiply it by the derivative of the inner function. In a formula, it is represented as \(f'(x) = g'(u(x)) \cdot u'(x)\), where \(g\) is the outer function and \(u\) is the inner function. This process breaks down the task into smaller, more manageable parts. Instead of attempting to differentiate the entire function at once, it focuses on each segment, one by one, which provides a clear path from the original function to its derivative.

Hyperbolic Sine

The hyperbolic sine function, denoted as \(\sinh(x)\), is an important mathematical function closely related to the familiar trigonometric sine function, yet it describes hyperbolic rather than circular relationships. It can be defined in terms of exponential functions: \(\sinh(x) = \frac{e^x - e^{-x}}{2}\). Differentiating \(\sinh(x)\) with respect to \(x\) gives us the hyperbolic cosine function, \(\cosh(x)\), which showcases a valuable interconnection between the two hyperbolic functions—much like the relationship between their trigonometric counterparts.

Hyperbolic Cosine

Similarly important is the hyperbolic cosine function, denoted as \(\cosh(x)\), which is the hyperbolic counterpart to the traditional cosine function. Defined using exponential functions, \(\cosh(x) = \frac{e^x + e^{-x}}{2}\), the relationship between \(\cosh(x)\) and \(\sinh(x)\) is apparent in their derivatives as well. In the context of differentiation, \(\cosh(x)\) appears as the derivative of \(\sinh(x)\). Just like with trigonometric functions, this connection simplifies the process of differentiation when dealing with combinations of hyperbolic functions.

Composite Function Differentiation

When faced with a composite function, like \(f(x) = \sinh(4x)\), where one function is nested inside another, we use the concept of composite function differentiation to find the derivative. This approach often incorporates the chain rule, pinpointing the inner and outer functions and treating them individually before combining their derivatives to yield the derivative of the composite function. By breaking down the differentiation process into steps, we can smoothly navigate through even the most intricate functions, separating and solving piece by piece and then weaving the results together for a comprehensive solution.