Problem 267
Question
$$ \text { If } y=(x+10)^{6}, \text { find }\left(\frac{d^{3} y}{d x^{3}}\right)_{x=2} $$
Step-by-Step Solution
Verified Answer
The third derivative of the function \(y = (x + 10)^{6}\) is \(\frac{d^3 y}{dx^3} = 120(x + 10)^3\). When evaluated at x = 2, \(\left(\frac{d^3 y}{dx^3}\right)_{x=2} = 207,360\).
1Step 1: Compute the first derivative with respect to x#
We are given the function \(y = (x + 10)^{6}\). To compute the first derivative with respect to x, we use the power rule of differentiation: \[\frac{d}{dx}(x^n) = nx^{n-1}\]
Therefore, the first derivative of y with respect to x is:
\[\frac{dy}{dx} = 6(x + 10)^{5}\]
2Step 2: Compute the second derivative with respect to x#
Now, we need to compute the second derivative of y with respect to x. To do this, we need to differentiate the first derivative with respect to x. We again use the power rule for differentiation:
\[\frac{d^2 y}{dx^2} = \frac{d}{dx} (6(x + 10)^{5}) = 6 \cdot 5 (x + 10)^{5-1} = 30(x + 10)^4\]
3Step 3: Compute the third derivative with respect to x#
Next, we compute the third derivative of y with respect to x by differentiating the second derivative with respect to x. Once more, we use the power rule of differentiation:
\[\frac{d^3 y}{dx^3} = \frac{d}{dx} (30(x + 10)^{4}) = 30 \cdot 4 (x + 10)^{4-1} = 120(x + 10)^3\]
4Step 4: Evaluate the third derivative at x = 2#
Finally, we are asked to find the value of the third derivative at x = 2. To do this, we simply substitute x = 2 into the expression for the third derivative:
\[\left(\frac{d^3 y}{dx^3}\right)_{x=2} = 120(2 + 10)^3 = 120 \cdot 12^3 = 120 \cdot 1728 = 207,360\]
The value of the third derivative of y with respect to x at x = 2 is 207,360.
Key Concepts
Power RuleHigher-order DerivativesEvaluation at a Point
Power Rule
The power rule is a fundamental concept in differential calculus. It provides a straightforward way to find the derivative of a function that is a power of a variable. This rule states that if you have a function of the form \(y = x^n\), where \(n\) is a constant, the derivative of \(y\) with respect to \(x\) is \(\frac{d}{dx}(x^n) = nx^{n-1}\).
In our given exercise, the function is \(y = (x + 10)^{6}\). To find the first derivative, we apply the power rule directly to obtain \(\frac{dy}{dx} = 6(x + 10)^{5}\). The power rule is useful because it simplifies finding derivatives of polynomial functions. Its simplicity and consistency make it a preferred method when faced with powers of variables. Remember to always multiply the given power by the original expression raised to one power less.
In our given exercise, the function is \(y = (x + 10)^{6}\). To find the first derivative, we apply the power rule directly to obtain \(\frac{dy}{dx} = 6(x + 10)^{5}\). The power rule is useful because it simplifies finding derivatives of polynomial functions. Its simplicity and consistency make it a preferred method when faced with powers of variables. Remember to always multiply the given power by the original expression raised to one power less.
Higher-order Derivatives
In calculus, higher-order derivatives refer to the process of differentiating a function multiple times. The first derivative represents the rate of change, like velocity in physics. However, higher-order derivatives can indicate further rates of change, like acceleration.
For the given function \(y = (x + 10)^{6}\), we start with the first derivative: \(\frac{dy}{dx} = 6(x + 10)^{5}\). We then find the second derivative by applying the power rule again to get \(\frac{d^2 y}{dx^2} = 30(x + 10)^{4}\). This process can be continued to find the third derivative: \(\frac{d^3 y}{dx^3} = 120(x + 10)^{3}\).
The notation \(\frac{d^n y}{dx^n}\) is used to indicate \(n\)-th order derivatives. Higher-order derivatives are valuable in various fields, such as physics and engineering, for understanding dynamics.
For the given function \(y = (x + 10)^{6}\), we start with the first derivative: \(\frac{dy}{dx} = 6(x + 10)^{5}\). We then find the second derivative by applying the power rule again to get \(\frac{d^2 y}{dx^2} = 30(x + 10)^{4}\). This process can be continued to find the third derivative: \(\frac{d^3 y}{dx^3} = 120(x + 10)^{3}\).
The notation \(\frac{d^n y}{dx^n}\) is used to indicate \(n\)-th order derivatives. Higher-order derivatives are valuable in various fields, such as physics and engineering, for understanding dynamics.
Evaluation at a Point
Once we have the derivative, we can evaluate it at a specific point to find the rate of change at that location. This process involves substituting a particular \(x\)-value into the derivative equation.
In the exercise, we needed to find the third derivative of \(y = (x + 10)^{6}\) evaluated at \(x = 2\). We obtained the third derivative: \(\frac{d^3 y}{dx^3} = 120(x + 10)^{3}\). By substituting \(x = 2\) into this expression, we computed \(120(2 + 10)^3 = 207,360\).
This step is crucial because it gives the exact rate of change at a specific \(x\) value, which can be particularly useful in practical applications. In real-world scenarios, evaluating derivatives at certain points can provide insights into the behavior of physical systems or changes occurring at precise moments.
In the exercise, we needed to find the third derivative of \(y = (x + 10)^{6}\) evaluated at \(x = 2\). We obtained the third derivative: \(\frac{d^3 y}{dx^3} = 120(x + 10)^{3}\). By substituting \(x = 2\) into this expression, we computed \(120(2 + 10)^3 = 207,360\).
This step is crucial because it gives the exact rate of change at a specific \(x\) value, which can be particularly useful in practical applications. In real-world scenarios, evaluating derivatives at certain points can provide insights into the behavior of physical systems or changes occurring at precise moments.
Other exercises in this chapter
Problem 264
$$ \text { If } y=\tan ^{-1} x, \text { find }\left(\frac{d^{2} y}{d x^{2}}\right)_{x=1} $$
View solution Problem 266
$$ \text { If } y=\cos ^{2} x, \text { find } \frac{d^{3} y}{d x^{3}} $$
View solution Problem 268
$$ \text { If } f(x)=\ln (\ln x), \text { find }\left(\frac{d^{3} y}{d x^{3}}\right)_{x=e} $$
View solution Problem 269
$$ \text { If } y=x^{3} \ln x, \text { find } \frac{d^{4} y}{d x^{4}} $$
View solution