Chapter 2: Problem 16
Compute the derivative of the given function. $$g(t)=10 t^{4}-\cos t+7 \sin t$$
Short Answer
Expert verified
The derivative of the function is \( g'(t) = 40t^3 + \sin t + 7\cos t \).
Step by step solution
01
Differentiate the Polynomial Term
To find the derivative of the polynomial term \[ 10t^4 \]use the power rule of differentiation, which states that \[ \frac{d}{dt} (t^n) = n t^{n-1} \].Apply this rule to get\[ \frac{d}{dt}(10t^4) = 10 \times 4t^{4-1} = 40t^3. \]
02
Differentiate the Cosine Function
Next, differentiate the cosine term \[ -\cos t \].The derivative of \( \cos t \) is \( -\sin t \), so applying this to \( -\cos t \), the derivative becomes\[ \frac{d}{dt}(-\cos t) = -(-\sin t) = \sin t. \]
03
Differentiate the Sine Function
Now, differentiate the sine term \[ 7\sin t \].The derivative of \( \sin t \)is \( \cos t \), so applying this rule, the derivative becomes\[ \frac{d}{dt}(7\sin t) = 7\cos t. \]
04
Sum the Derivatives
Combine the derivatives from Steps 1, 2, and 3 to calculate the derivative of the entire function\[ g'(t) = 40t^3 + \sin t + 7\cos t. \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Power Rule
Understanding the power rule is fundamental in calculus, especially when working with polynomial functions. It provides a straightforward method for differentiating terms like \( t^n \), where \( n \) is a real number. This rule states that when differentiating \( t^n \), you multiply by the exponent \( n \) and then subtract one from the exponent to find its new power.
For example, evaluating \( \frac{d}{dt}(t^n) \) yields \( nt^{n-1} \). This transformation makes complex polynomial differentiation much more manageable. It's a cornerstone technique applied in countless calculus problems.
In our exercise, applying the power rule to \( 10t^4 \) simplifies the process to \( 40t^3 \), making calculations and understanding easier. Utilizing the power rule efficiently helps you develop proficiency in calculus.
For example, evaluating \( \frac{d}{dt}(t^n) \) yields \( nt^{n-1} \). This transformation makes complex polynomial differentiation much more manageable. It's a cornerstone technique applied in countless calculus problems.
In our exercise, applying the power rule to \( 10t^4 \) simplifies the process to \( 40t^3 \), making calculations and understanding easier. Utilizing the power rule efficiently helps you develop proficiency in calculus.
Differentiation of Trigonometric Functions
Differentiating trigonometric functions plays a crucial role in understanding calculus. Trig functions include sine, cosine, tangent, and others, each having specific derivatives.
The core idea is that each standard trig function has a set derivative: for \( \sin t \), it's \( \cos t \), and for \( \cos t \), it's \(-\sin t \). These simple rules help us tackle more complex problems involving trig functions within equations.
When working on calculus problems with trig elements, always keep these fundamental derivatives in mind. It will streamline your differentiation process and let you focus on the broader problem-solving strategy.
The core idea is that each standard trig function has a set derivative: for \( \sin t \), it's \( \cos t \), and for \( \cos t \), it's \(-\sin t \). These simple rules help us tackle more complex problems involving trig functions within equations.
When working on calculus problems with trig elements, always keep these fundamental derivatives in mind. It will streamline your differentiation process and let you focus on the broader problem-solving strategy.
Polynomial Differentiation
Polynomial differentiation involves finding the derivative of each term within a polynomial. This approach allows you to handle terms independently using rules like the power rule.
This process is straightforward: treat each term in isolation and apply differentiation rules. Then, combine the results to get the derivative of the entire polynomial function.
In our exercise, the function is \( 10t^4 - \cos t + 7\sin t \). We differentiated each part independently: the polynomial term separately from the trigonometric terms. Polynomial differentiation becomes intuitive and straightforward when you understand and apply the appropriate rules.
This process is straightforward: treat each term in isolation and apply differentiation rules. Then, combine the results to get the derivative of the entire polynomial function.
In our exercise, the function is \( 10t^4 - \cos t + 7\sin t \). We differentiated each part independently: the polynomial term separately from the trigonometric terms. Polynomial differentiation becomes intuitive and straightforward when you understand and apply the appropriate rules.
Sine Function
The sine function, \( \sin t \), is one of the foundational trigonometric functions in calculus. Its derivative is \( \cos t \), a result that is frequently used in calculus-based problems.
Familiarity with this relationship aids in smooth computation and analysis of calculus problems. Recognizing that the sine function’s derivative is the cosine function allows for seamless transition between differentiation steps, particularly in composite functions or equations.
In scenarios where sine terms appear, this derivative relationship helps streamline your process and focuses your work on broader calculus concepts.
Familiarity with this relationship aids in smooth computation and analysis of calculus problems. Recognizing that the sine function’s derivative is the cosine function allows for seamless transition between differentiation steps, particularly in composite functions or equations.
In scenarios where sine terms appear, this derivative relationship helps streamline your process and focuses your work on broader calculus concepts.
Cosine Function
The cosine function, \( \cos t \), features prominently in calculus due to its regular appearance in trigonometric problems. Its derivative, \(-\sin t\), is essential in analyzing changes and slopes of trigonometric graphs.
This knowledge is crucial for handling any calculus problem involving the differentiation of trig functions. Knowing how the cosine’s derivative relates to the sine function allows you to predict changes in related functions seamlessly.
In this guide, the cosine derivative was applied swiftly to solve a calculus problem, demonstrating its direct and critical application in derivative calculus.
This knowledge is crucial for handling any calculus problem involving the differentiation of trig functions. Knowing how the cosine’s derivative relates to the sine function allows you to predict changes in related functions seamlessly.
In this guide, the cosine derivative was applied swiftly to solve a calculus problem, demonstrating its direct and critical application in derivative calculus.
