Problem 18
Question
Find \(D_{x} y\) using the rules of this section. $$ y=2 x^{-6}+x^{-1} $$
Step-by-Step Solution
Verified Answer
The derivative is \(D_{x} y = -12x^{-7} - x^{-2}\).
1Step 1: Identify the Derivative Rule
To differentiate the function \( y = 2x^{-6} + x^{-1} \), we will use the power rule for derivatives. The power rule states that \( \frac{d}{dx}[x^n] = nx^{n-1} \).
2Step 2: Differentiate Each Term
Apply the power rule to each term individually. - For the term \(2x^{-6}\), its derivative is \(-12x^{-7}\) because \(-6 \cdot 2 = -12\) and subtracting one from the exponent \(-6\) gives \(-7\).- For the term \(x^{-1}\), its derivative is \(-1x^{-2}\) because \(-1 \cdot 1 = -1\) and subtracting one from the exponent \(-1\) gives \(-2\).
3Step 3: Combine the Results
Combine the derivatives of each term to find the derivative of the entire expression. So, \(D_{x} y = -12x^{-7} - x^{-2}\).
Key Concepts
Power Rule for DerivativesDifferentiation TechniquesCalculus Problem Solving
Power Rule for Derivatives
When dealing with polynomial functions in calculus, one of the most powerful and straightforward rules for finding derivatives is the power rule. The power rule is a specific differentiation technique designed to simplify finding the derivative of monomials—expressions that feature a single term with a variable raised to an exponent. Mathematically, if you have a function in the form of \( y = x^n \), the power rule tells us that the derivative of that function with respect to \( x \), noted as \( \frac{d}{dx}[x^n] \), is \( nx^{n-1} \).
This means that you simply take the exponent, \( n \), multiply it by the coefficient in front of \( x \), and then reduce the exponent by one. Each monomial can be treated independently when applying the power rule, making it highly efficient for more complex polynomial expressions. The simplicity of this rule makes it one of the first and most easily grasped techniques learned by calculus students.
This means that you simply take the exponent, \( n \), multiply it by the coefficient in front of \( x \), and then reduce the exponent by one. Each monomial can be treated independently when applying the power rule, making it highly efficient for more complex polynomial expressions. The simplicity of this rule makes it one of the first and most easily grasped techniques learned by calculus students.
Differentiation Techniques
Differentiation techniques are various strategies that mathematicians and scientists use to find the rate of change of a function. The power rule is just one of several differentiation techniques taught in calculus. Each technique has its specific use case and efficiency depending on the type of function at hand.
For the function \( y=2x^{-6}+x^{-1} \), the power rule was perfectly suited. By applying the power rule to each term individually:
These methods must be applied to each term of a polynomial to find the complete derivative of the function. Thus, having a command of the different techniques and when to use them is crucial in calculus.
For the function \( y=2x^{-6}+x^{-1} \), the power rule was perfectly suited. By applying the power rule to each term individually:
- For \( 2x^{-6} \), the derivative becomes \( -12x^{-7} \). We achieve this by multiplying \(-6\) with \( 2 \) and dropping the exponent by one (\(-6 - 1 = -7\)).
- For \( x^{-1} \), the derivative turns into \( -1x^{-2} \). Here, \( -1 \) from the exponent multiplies by the coefficient of \( 1 \), and we subtract one from the exponent \(-1\), resulting in \(-2\).
These methods must be applied to each term of a polynomial to find the complete derivative of the function. Thus, having a command of the different techniques and when to use them is crucial in calculus.
Calculus Problem Solving
Calculus problem solving often involves breaking down complex expressions into simpler parts and then applying known rules and techniques to find solutions. Practicing calculus problem solving helps students to develop logical thinking and the ability to tackle challenging problems by identifying patterns and choosing appropriate techniques.
In the exercise presented, solving for \( D_x y \) involved breaking down the function \( y=2x^{-6}+x^{-1} \) and applying the power rule to each term separately. This step-by-step decomposition simplifies the process and highlights the importance of understanding each part of the equation individually before combining the results.
In the exercise presented, solving for \( D_x y \) involved breaking down the function \( y=2x^{-6}+x^{-1} \) and applying the power rule to each term separately. This step-by-step decomposition simplifies the process and highlights the importance of understanding each part of the equation individually before combining the results.
- Apply differentiation rules, such as the power rule, to decompose the problem.
- Identify each polynomial term and find its individual derivative.
- Finally, combine the individual derivatives to solve the complete problem.
Other exercises in this chapter
Problem 18
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$$ \underline{\phantom{xxx}} \text { find } D_{x} y . $$ $$ y=\sec ^{3} x $$
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Use \(f^{\prime}(x)=\lim _{h \rightarrow 0}[f(x+h)-f(x)] / h\) to find the derivative at \(x\). $$ G(x)=\frac{2 x}{x^{2}-x} $$
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