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Chapter 2: Problem 7

Find the derivative of \(h(t)=t \sin t+\cos t\) \(h^{\prime}(t)=\) _______

Short Answer

Expert verified
\[ h'(t) = t \cos t \]

Step by step solution

01

Apply the Sum Rule

The function given is the sum of two functions: \[ h(t) = t \sin t + \cos t \] According to the sum rule for differentiation, the derivative of a sum is the sum of the derivatives. Hence \[ h'(t) = (t \sin t)' + (\cos t)' \]
02

Differentiate the First Term

To differentiate the term \( t \sin t \), apply the product rule. The product rule states: \[ (uv)' = u'v + uv' \] Here, let \( u = t \) and \( v = \sin t \), so we get: \[ (t \sin t)' = (t)' (\sin t) + (t) (\sin t)' \] Now, \( (t)' = 1 \) and \( (\sin t)' = \cos t \). Thus, we have: \[ (t \sin t)' = 1 \cdot \sin t + t \cdot \cos t = \sin t + t \cos t \]
03

Differentiate the Second Term

Differentiate the term \( \cos t \). Using the derivative rule for trigonometric functions, we know: \[ (\cos t)' = -\sin t \]
04

Combine the Results

Now, sum the derivatives of the individual terms as per the sum rule: \[ h'(t) = (t \sin t)' + (\cos t)' = (\sin t + t \cos t) + (-\sin t) \] Simplify the expression: \[ h'(t) = \sin t + t \cos t - \sin t \] Notice that \( \sin t \) and \( -\sin t \) cancel each other: \[ h'(t) = t \cos t \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum Rule in Differentiation
The sum rule is a fundamental principle in differentiation. It states that the derivative of the sum of two functions is the sum of their individual derivatives.
This can be written as: \[ (f(x) + g(x))' = f'(x) + g'(x) \]
In our exercise, we have the function \ h(t) = t \sin t + \cos t \.
Applying the sum rule, we find the derivative by splitting the problem:
\ h'(t) = (t \sin t)' + (\cos t)'\.
This allows us to solve each part separately, simplifying our task.
Understanding the Product Rule
The product rule is essential when differentiating products of two functions.
It states that the derivative of the product of two functions is given by:
\[ (uv)' = u'v + uv' \]
In our exercise, for \( t \sin t \), consider \ u = t \ and \ v = \sin t \.
Applying the product rule, we get:
\[ (t \sin t)' = (t)' \sin t + t (\sin t)' = 1 \sin t + t \cos t = \sin t + t \cos t \]
Here, we first find each derivative separately and then sum them up.
Differentiation of Trigonometric Functions
Differentiating trigonometric functions follows specific rules:
\[ (\sin t)' = \cos t \]
\[ (\cos t)' = -\sin t \]
For our given function \( \cos t \), we apply the rule directly:
\[(\cos t)' = -\sin t \]
By combining this with the sum and product rule results, we get the final derivative:
\[ h'(t) = (\sin t + t \cos t) + (-\sin t) = t \cos t \]
Thus, the final result is \[ h'(t) = t \cos t \].

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Most popular questions from this chapter

Let \(f(x)=\frac{\tan (x)-5}{\sec (x)} .\) Find the following: $$ \begin{array}{l} \text { 1. } f^{\prime}(x)= \\ \text { 2. } f^{\prime}(1)= \end{array} $$

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