Question

Find the derivatives of the following functions. $$f(x)=-\sinh ^{3} 4 x$$

Short answer

Answer: The derivative of the function $$f(x)=-\sinh ^{3} 4x$$ is $$f'(x)=-12\cosh(4x)\sinh^2(4x)$$.

Step-by-step solution

Break down the function

We can write the given function as follows: $$f(x) = -(\sinh(4x))^3$$ So, we have an outer function $$g(u)=-u^3$$ and an inner function $$u = h(x)=\sinh(4x)$$. Now we need to find the derivatives of both functions with respect to $$x$$.

Find the derivative of the outer function

We need to differentiate $$g(u)=-u^3$$ with respect to $$u$$. Using the power rule, we get: $$g'(u)=-3u^2$$

Find the derivative of the inner function

We need to differentiate $$h(x)=\sinh(4x)$$ with respect to $$x$$. Using the chain rule, we get: $$h'(x)=\cosh(4x)\cdot 4$$

Apply the chain rule

Now, we can use the chain rule to find the derivative of the given function with respect to $$x$$. According to the chain rule, $$f'(x)=g'(h(x))\cdot h'(x)$$ So, plugging in the expressions from Steps 2 and 3, we get: $$f'(x)=-3(\sinh(4x))^2\cdot\cosh(4x)\cdot4$$

Simplify the result

Finally, let's simplify the expression: $$f'(x)=-12\cosh(4x)\sinh^2(4x)$$ So, the derivative of the given function is: $$f'(x)=-12\cosh(4x)\sinh^2(4x)$$

Key concepts

Chain Rule

The chain rule is a cornerstone of calculus, essential when dealing with composite functions - functions made up of two or more combined functions. Imagine stacking two processes: one follows the other. To find the rate of change for the overall process, the chain rule instructs us to differentiate each function separately and multiply the results.

For a composite function represented as \( f(g(x)) \), the derivative \( f'(x) \) is found by multiplying the derivative of the outer function \( f'(g(x)) \) by the derivative of the inner function \( g'(x) \). This rule allows us to dissect complex expressions and tackle them piece by piece, simplifying the differentiation process.

Hyperbolic Functions

Hyperbolic functions, similar to trigonometric functions, are important in many areas of mathematics including calculus. The hyperbolic sine \( \sinh(x) \) and hyperbolic cosine \( \cosh(x) \) are two fundamental hyperbolic functions.

They're defined using exponential functions as:
\[ \sinh(x) = \frac{e^x - e^{-x}}{2} \] and
\[ \cosh(x) = \frac{e^x + e^{-x}}{2} \]
This creates a rich connection to exponential growth and decay processes, making them incredibly useful for modeling real-world phenomena like the shape of a hanging cable or the distribution of heat over time.

Power Rule

The power rule is a straightforward but powerful tool in calculus, used to differentiate functions of the form \( f(x) = x^n \) where \( n \) is any real number. The rule states that the derivative of \( f(x) \) with respect to \( x \) is \( n \) times the function with its power reduced by one:
\[ f'(x) = nx^{n-1} \]
Applying the power rule is like turning down the volume on a function - you decrease its power while amplifying the coefficient. This rule allows rapid computations and is especially handy when dealing with polynomial functions.

Derivative of sinh

Considering \( \sinh(x) \) as a hyperbolic function, its derivative holds a neat relationship with another hyperbolic function. The derivative of \( \sinh(x) \) is \( \cosh(x) \), the hyperbolic cosine of \( x \).

If we have \( \sinh(kx) \) for some constant \( k \), we must remember to multiply by \( k \) due to the chain rule, as seen in \( \sinh(4x) \). So, the derivative of \( \sinh(4x) \) is \( 4 \cosh(4x) \). This captures the essence of calculus - finding patterns and connections which simplify the process of finding rates of change.