Question

Apply the Chain Rule more than once to find the indicated derivative. \(D_{t}\left[\sin ^{3}(\cos t)\right]\)

Short answer

The derivative is \(-3\sin^2(\cos t) \cdot \cos(\cos t) \cdot \sin t\)."

Step-by-step solution

Identify the Outer Function

The expression is given as \(\sin^3(\cos t)\). The outermost function here is \(u^3\), where \(u = \sin(\cos t)\). We will first differentiate this with respect to the inner function \(u\).

Differentiate the Outer Function

Differentiate the function \(u^3\) using the power rule. The derivative of \(u^3\) with respect to \(u\) is \(3u^2\). Thus, the derivative of \(\sin^3(\cos t)\) with respect to \(\sin(\cos t)\) is \(3\sin^2(\cos t)\).

Identify the Middle Function

Now, move to the next inner layer, \(\sin(\cos t)\). The function here can be seen as \(\sin(v)\) where \(v = \cos t\). We will now differentiate this function with respect to \(v\).

Differentiate the Middle Function

The derivative of \(\sin(v)\) with respect to \(v\) is \(\cos(v)\). Therefore, the derivative of \(\sin(\cos t)\) with respect to \(\cos t\) is \(\cos(\cos t)\).

Identify the Innermost Function

The innermost function is \(\cos t\) with respect to \(t\). Now, we need to differentiate \(\cos t\) with respect to \(t\).

Differentiate the Innermost Function

The derivative of \(\cos(t)\) with respect to \(t\) is \(-\sin(t)\).

Apply the Chain Rule

Now combine all the derivatives using the chain rule. The derivative of the entire function \(D_t[\sin^3(\cos t)]\) is calculated as:- First, multiply the derivative of the outer function by the derivative of the middle function, resulting in \(3\sin^2(\cos t) \cdot \cos(\cos t)\).- Then, multiply this result by the derivative of the innermost function (i.e. \(-\sin t\)).Finally, the complete derivative is \(-3\sin^2(\cos t) \cdot \cos(\cos t) \cdot \sin t\).

Key concepts

Differentiation

In calculus, differentiation is a fundamental concept that involves finding the derivative of a function. A derivative represents the rate at which a function is changing at any given point, essentially providing the slope of the tangent line to the function's graph. Understanding how to differentiate functions allows you to analyze how quantities change in relation to others.

The process of differentiation uses specific rules to determine how the derivative of a function behaves. These rules simplify the process especially when dealing with more complex functions. Some of the common rules include:

• Power Rule: The derivative of any power function, such as \(x^n\), is \(nx^{n-1}\).
• Constant Rule: The derivative of a constant is zero.
• Sum Rule: The derivative of a sum of functions is the sum of their derivatives.

Differentiation is crucial because it helps in understanding and interpreting the behavior of functions in various real-world scenarios. Whether tracking speed, growth, or declines, knowing how to differentiate effectively empowers you to gain deeper insights into problems.

Composite Functions

Composite functions involve the combination of two different functions, where one function is applied to the result of another. This nesting of functions is expressed as \(f(g(x))\). When differentiating composite functions, it is essential to understand how these functions are related.

The Chain Rule is the primary strategy used for differentiating composite functions. This rule states that the derivative of \(f(g(x))\) is \(f'(g(x)) \cdot g'(x)\). In simpler terms, differentiate the outer function, leave the inner function unchanged, and multiply by the derivative of the inner function.

Composite functions frequently appear in calculus, and mastering the Chain Rule is necessary to untangle these layers of functions. Practicing the Chain Rule in varying scenarios enhances your ability to handle complex expressions and equations efficiently.

Trigonometric Derivatives

Trigonometric derivatives are formulas that provide the rate of change for trigonometric functions, such as sine, cosine, and tangent. These derivatives are pivotal when dealing with periodic functions and oscillations, like sound waves or circular motions.

The basic trigonometric derivatives include:

• Derivative of Sine: The derivative of \( \sin(x) \) is \( \cos(x) \).
• Derivative of Cosine: The derivative of \( \cos(x) \) is \( -\sin(x) \).
• Derivative of Tangent: The derivative of \( \tan(x) \) is \( \sec^2(x) \).

Understanding these basic derivatives simplifies the differentiation of more complex trigonometric expressions. They'll often be combined with other rules, like the Chain Rule, to tackle intricate calculus problems.

Trigonometric derivatives help in modeling and solving real-world phenomena wherever wave patterns or rotational dynamics are involved, allowing for deep exploration and comprehension of periodic behavior.