Question

Given \(x=\ln \left(u^{2}-v^{2}\right), u=t^{2}, v=\cos t\), find \(d x / d t\).

Short answer

\(\frac{dx}{dt} = \frac{4t^{3} + 2\cos t \sin t}{t^{4} - \cos^{2} t}\)

Step-by-step solution

Identify Given Functions

Given the functions: \(x = \ln(u^{2} - v^{2})\), \(u = t^{2}\), \(v = \cos t\). Notice that we need to find \(\frac{dx}{dt}\).

Express x in Terms of t

Rewrite \(x\) in terms of \(t\) using the given substitutions: \(u = t^{2}\) \(v = \cos t\)Thus, \(x = \ln((t^{2})^{2} - (\cos t)^{2})\)\(x = \ln(t^{4} - \cos^{2} t)\)

Use the Chain Rule

To differentiate \(x\) with respect to \(t\), apply the chain rule. First, find the derivative of the natural logarithm function: \(\frac{d}{dt}[\ln(t^{4} - \cos^{2} t)] = \frac{1}{t^{4} - \cos^{2} t} \cdot \frac{d}{dt}(t^{4} - \cos^{2} t)\)

Differentiate the Inner Function

Differentiate the inner function \(t^{4} - \cos^{2} t\) with respect to \(t\): \(\frac{d}{dt}(t^{4} - \cos^{2} t) = \frac{d}{dt}(t^{4}) - \frac{d}{dt}(\cos^{2} t)\)Using the power rule: \(\frac{d}{dt}(t^{4}) = 4t^{3}\)Using the chain rule for \(\cos^{2} t\): \(\frac{d}{dt}(\cos^{2} t) = \frac{d}{d\cos t}(\cos^{2} t) \cdot \frac{d}{dt}(\cos t)\)\(= 2\cos t \cdot (-\sin t) = -2\cos t \sin t\)Therefore, \(\frac{d}{dt}(t^{4} - \cos^{2} t) = 4t^{3} + 2\cos t \sin t\)

Combine the Results

Combine the previous results to find \(\frac{dx}{dt}\): \(\frac{dx}{dt} = \frac{1}{t^{4} - \cos^{2} t} \cdot (4t^{3} + 2\cos t \sin t)\)Simplify if possible.

Key concepts

Chain Rule

The Chain Rule is a fundamental technique in calculus used for differentiating compositions of functions. When you need to find the derivative of a composition, you differentiate the outer function first and then multiply it by the derivative of the inner function.
For example, if you have a function composed like \text{f(g(t))}\, the chain rule states:
\(\frac{d}{dt} [f(g(t))] = f'(g(t)) \times g'(t)\)
In our exercise, where we have \(x = \ln(u^{2} - v^{2})\), we need to differentiate the natural logarithm, which is the outer function, and then multiply it by the derivative of the inner function \(u^{2} - v^{2}\).
Let's go through each step to get a better understanding:

• Identify the outer and inner function.
• Differentiate the outer function.
• Differentiate the inner function.
• Multiply them together as per the chain rule.

This step-by-step logic is crucial when dealing with more complex functions, as it breaks down the problem into manageable parts.

Natural Logarithm Differentiation

Differentiating the natural logarithm function \(\text{ln}(x)\) requires understanding its basic rule. The derivative of \(\text{ln}(x)\) is:
\(\frac{d}{dx}[\text{ln}(x)] = \frac{1}{x}\)
In the given problem, we deal with \(\text{ln}(u^2 - v^2)\)
To apply the chain rule, first find the derivative of the logarithm itself:

• The outer function is \(\text{ln}(x)\), whose derivative is \(\frac{1}{u^2 - v^2}\).
• We then multiply this result by the derivative of \(u^2 - v^2\).

Combining these steps gives us the derivative of \(\text{ln}(u^2 - v^2)\), which simplifies the differentiation process and helps us move closer to the solution.

Power Rule

The Power Rule simplifies differentiation when dealing with polynomials. It states that if you have a term of the form \(t^n\), its derivative is:
\(\frac{d}{dt}[t^n] = n t^{n-1} \).
In the given problem, we see the term \(t^4\):
Using the power rule: \(\frac{d}{dt}[t^4] = 4t^3\).
Another part of the exercise involves \(\text{cos}^2{t}\), and although it appears slightly different, the chain rule transforms it in similar steps:

• First differentiate \(\text{cos}^2{t}\) using the chain rule: \(\frac{d}{dt}{[\text{cos}^2(t)]} = 2\text{cos}(t) (-\text{sin}(t)) = -2\text{cos}(t) \text{sin}(t)\).

Combining both results of the differentiated terms and applying them correctly ensures complete and correct differentiation. Breaking down big problems into these basic rules makes complex derivatives more manageable.