Question

Using the Quotient Rule In Exercises \(7-12,\) use the Quotient Rule to find the derivative of the function. $$ f(t)=\frac{\cos t}{t^{3}} $$

Short answer

The derivative of the function \(f(t) = \frac{\cos t}{t^{3}}\) is \(f'(t) = \frac{-t^{3}\sin t - 3t^{2}\cos t}{t^{6}}\).

Step-by-step solution

Identify the functions u and v

In the given function \(f(t) = \frac{\cos t}{t^{3}}\), identify the numerator as function \(u=\cos t\) and the denominator as function \(v=t^{3}\).

Calculate the derivatives of u and v

Calculate the derivatives of \(u\) and \(v\). The derivative of \(u=\cos t\) is \(u'=-\sin t\). The derivative of \(v=t^3\) is \(v'=3t^2\).

Apply the Quotient Rule

Apply the Quotient Rule to find the derivative of the function \(f(t) = \frac{\cos t}{t^{3}}\), which is \(f'(t) = \frac{vu' - uv'}{v^2} = \frac{t^{3}(-\sin t) - \cos t(3t^{2}) }{(t^3)^2} = \frac{-t^{3}\sin t - 3t^{2}\cos t}{t^{6}}\). You may simplify it further if needed.

Key concepts

Derivative of a Function

Understanding the derivative of a function is essential in calculus. This concept is often used to determine the rate at which something changes. To grasp this fundamental idea, imagine you are driving a car and your speedometer is broken. To calculate your speed, you could measure how much distance you cover over a certain time period. If you go 100 meters in 2 seconds, you could say your 'average speed' was 50 meters per second during that time. Derivatives work similarly, but instead of average speed, they find the 'instantaneous speed' or the exact rate of change at a single point in time.

For a mathematical function, we represent this 'instantaneous speed' as the slope of the function's graph at a certain point. When we calculate a derivative, we are finding the slope of the tangent line to the graph of the function at any point. This process is often symbolized as \( f'(x) \), which denotes the derivative of the function \( f(x) \). The derivative can provide vital information about a function, such as where it increases or decreases, and where it reaches its highest and lowest points.

Derivatives of Trigonometric Functions

Trigonometric functions like \( \sin(x) \) and \( \cos(x) \) describe various relationships within right-angled triangles and are also related to the unit circle. They are widely used in physics, engineering, and mathematics. When it comes to taking derivatives of these functions, there are specific rules that apply. For instance, the derivative of \( \sin(x) \) is \( \cos(x) \) and the derivative of \( \cos(x) \) is \( -\sin(x) \). It's fascinating to note that the derivatives of trigonometric functions are cyclic - differentiating \( \sin \) turns into \( \cos \) and differentiating \( \cos \) turns into \( -\sin \), and if you differentiate \( -\sin \) you get \( -\cos \) and so on.

Knowing these derivatives is crucial when dealing with more complex functions involving trigonometric expressions, especially when these functions are part of a quotient where you're applying additional rules for differentiation.

Applying the Quotient Rule

In calculus, the Quotient Rule is a method for finding the derivative of a function that is the ratio of two differentiable functions. When you have a function represented by \( f(x) = \frac{u(x)}{v(x)} \) where both \( u(x) \) and \( v(x) \) are themselves differentiable functions, the Quotient Rule can be applied. The Quotient Rule states that the derivative of \( f(x) \) is \( f'(x) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2} \). Breaking this down, \( u'(x) \) and \( v'(x) \) are the derivatives of \( u(x) \) and \( v(x) \) respectively.

It's important to carefully follow the subtraction in the numerator to avoid any sign errors. Also, understanding how to simplify complex fractions can be quite helpful after applying the Quotient Rule, as it often results in an expression that may require further reduction. Finally, this rule is a powerful tool when combined with the understanding of derivatives of trigonometric functions, allowing for complex expressions to be differentiated with ease.