Question

Compute the derivative of the given function. $$h(t)=\sin ^{4}(2 t)$$

Short answer

The derivative of \( h(t) = \sin^4(2t) \) is \( 8\sin^3(2t)\cos(2t) \).

Step-by-step solution

Understanding the Function

We need to find the derivative of the function \( h(t) = \sin^4(2t) \). This function represents \( (\sin(2t))^4 \), which means \( \sin(2t) \) is raised to the fourth power.

Apply the Chain Rule

To differentiate \( h(t) = (\sin(2t))^4 \), we use the chain rule, which is a formula to compute the derivative of a composite function. Identify \( u(t) = \sin(2t) \) and rewrite the function as \( h(t) = u^4(t) \).

Differentiate the Outer Function

First, differentiate the outer function \( u^4 \). The derivative is \( 4u^3 \) by using the power rule. Thus, \( \frac{d}{du}[u^4] = 4u^3 \).

Differentiate the Inner Function

Next, differentiate the inner function \( u(t) = \sin(2t) \) to find \( \frac{du}{dt} \). Use the chain rule again; differentiate \( \sin(x) \) to get \( \cos(x) \), and the derivative of \( 2t \) is \( 2 \). So, \( \frac{du}{dt} = 2\cos(2t) \).

Combine the Derivatives

Combine the derivatives using the chain rule: \( \frac{dh}{dt} = \frac{dh}{du} \cdot \frac{du}{dt} = 4u^3 \cdot 2\cos(2t) \). Substitute back \( u = \sin(2t) \) to get \( \frac{dh}{dt} = 8(\sin(2t))^3 \cos(2t) \).

Key concepts

Differentiation

Differentiation is a fundamental concept in calculus, crucial for understanding how functions change. It allows us to find the rate at which a function is changing at any given point, which is essential in fields like physics, engineering, and economics. The derivative of a function measures its instantaneous rate of change, similar to how speed tells us how fast an object moves.

• A derivative is often denoted by \( \frac{dy}{dx} \), which represents the change in \( y \) with respect to \( x \).
• The process of finding a derivative is known as differentiation.
• Understanding differentiation helps in solving various types of problems, including optimization and finding tangent lines to curves.

When we differentiate a composite function, like in our exercise, we use specific rules to find the derivative efficiently. By applying the right techniques, we can tackle even complex functions with ease.

Power Rule

The power rule is a quick method for differentiating functions of the form \( x^n \) where \( n \) is a constant. It's one of the simplest but most powerful tools in calculus.

• According to the power rule, the derivative of \( x^n \) is \( nx^{n-1} \).
• This rule helps in breaking down more complex functions into manageable parts for differentiation.
• In the step-by-step solution, when we found the derivative of \( u^4 \), we used the power rule to get \( 4u^3 \).

The power rule shines when used in conjunction with other rules, such as the chain rule, to simplify the process of differentiation. By combining these methods, finding derivatives becomes straightforward, even for functions raised to a power, as seen in the problem \( h(t) = \sin^4(2t) \).

Trigonometric Functions

Trigonometric functions are vital in calculus due to their periodic nature, which models oscillating phenomena like sound waves, light, and tides. Differentiating trigonometric functions is a skill necessary for understanding their behavior over time.

• The derivative of \( \sin(x) \) is \( \cos(x) \) and that of \( \cos(x) \) is \(-\sin(x) \).
• Using these derivatives helps us understand how changes in angles impact the function's output.
• In our problem, we found the derivative of \( \sin(2t) \) by recognizing it as a chain of \( \sin(x) \) and \( 2t \).

Understanding how to differentiate trigonometric functions is crucial, especially when combined with other derivatives like power and chain rules. It ensures you can handle functions involving trigonometric expressions, such as in the expression \( \sin^4(2t) \).