Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Discover in-depth answers to your questions from a wide network of experts on our user-friendly Q&A platform. Get quick and reliable solutions to your questions from a community of experienced experts on our platform.
Sagot :
To solve the problem of finding the derivative [tex]\(\frac{dy}{dx}\)[/tex] using the chain rule, given the functions [tex]\( y = f(u) \)[/tex] and [tex]\( u = g(x) \)[/tex], specifically:
[tex]\[ y = 5u^4 - 5 \quad \text{and} \quad u = 3\sqrt{x} \][/tex]
we can break the problem down into the following steps:
1. Find [tex]\(\frac{dy}{du}\)[/tex]:
Begin by differentiating [tex]\( y \)[/tex] with respect to [tex]\( u \)[/tex]. The function [tex]\( y = 5u^4 - 5 \)[/tex] is a polynomial in terms of [tex]\( u \)[/tex].
[tex]\[ \frac{dy}{du} = \frac{d(5u^4 - 5)}{du} \][/tex]
Using the power rule of differentiation, which states that [tex]\(\frac{d}{dx}[x^n] = nx^{n-1}\)[/tex], we get:
[tex]\[ \frac{dy}{du} = 5 \cdot 4u^{3} = 20u^3 \][/tex]
2. Find [tex]\(\frac{du}{dx}\)[/tex]:
Next, we differentiate [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex]. The function [tex]\( u = 3\sqrt{x} \)[/tex] can also be written as [tex]\( u = 3x^{1/2} \)[/tex].
[tex]\[ \frac{du}{dx} = \frac{d(3x^{1/2})}{dx} \][/tex]
Again, using the power rule:
[tex]\[ \frac{du}{dx} = 3 \cdot \frac{1}{2} x^{-1/2} = \frac{3}{2} x^{-1/2} \][/tex]
Which can be simplified to:
[tex]\[ \frac{du}{dx} = \frac{3}{2\sqrt{x}} \][/tex]
3. Apply the chain rule:
The chain rule in Leibniz's notation states that:
[tex]\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \][/tex]
Substitute the expressions we found for [tex]\(\frac{dy}{du}\)[/tex] and [tex]\(\frac{du}{dx}\)[/tex] into this formula:
[tex]\[ \frac{dy}{dx} = (20u^3) \cdot \left(\frac{3}{2\sqrt{x}}\right) \][/tex]
4. Substitute [tex]\( u = 3\sqrt{x} \)[/tex] into the expression:
Since [tex]\( u = 3\sqrt{x} \)[/tex], we can substitute this back into our expression for [tex]\( \frac{dy}{dx} \)[/tex]:
[tex]\[ u = 3\sqrt{x} \implies u^3 = (3\sqrt{x})^3 = 27x^{3/2} \][/tex]
Hence:
[tex]\[ \frac{dy}{dx} = 20(27x^{3/2}) \cdot \left(\frac{3}{2\sqrt{x}}\right) \][/tex]
Simplify this expression by multiplying the coefficients and combining the terms:
[tex]\[ \frac{dy}{dx} = 20 \cdot 27x^{3/2} \cdot \frac{3}{2\sqrt{x}} \][/tex]
Combine the terms:
[tex]\[ \frac{dy}{dx} = \frac{20 \cdot 27 \cdot 3}{2} x^{(3/2) - (1/2)} \][/tex]
[tex]\[ \frac{dy}{dx} = \frac{1620}{2} x^{1} \][/tex]
[tex]\[ \frac{dy}{dx} = 810x \][/tex]
Thus, the derivative [tex]\(\frac{dy}{dx}\)[/tex] is:
[tex]\[ \boxed{810x} \][/tex]
[tex]\[ y = 5u^4 - 5 \quad \text{and} \quad u = 3\sqrt{x} \][/tex]
we can break the problem down into the following steps:
1. Find [tex]\(\frac{dy}{du}\)[/tex]:
Begin by differentiating [tex]\( y \)[/tex] with respect to [tex]\( u \)[/tex]. The function [tex]\( y = 5u^4 - 5 \)[/tex] is a polynomial in terms of [tex]\( u \)[/tex].
[tex]\[ \frac{dy}{du} = \frac{d(5u^4 - 5)}{du} \][/tex]
Using the power rule of differentiation, which states that [tex]\(\frac{d}{dx}[x^n] = nx^{n-1}\)[/tex], we get:
[tex]\[ \frac{dy}{du} = 5 \cdot 4u^{3} = 20u^3 \][/tex]
2. Find [tex]\(\frac{du}{dx}\)[/tex]:
Next, we differentiate [tex]\( u \)[/tex] with respect to [tex]\( x \)[/tex]. The function [tex]\( u = 3\sqrt{x} \)[/tex] can also be written as [tex]\( u = 3x^{1/2} \)[/tex].
[tex]\[ \frac{du}{dx} = \frac{d(3x^{1/2})}{dx} \][/tex]
Again, using the power rule:
[tex]\[ \frac{du}{dx} = 3 \cdot \frac{1}{2} x^{-1/2} = \frac{3}{2} x^{-1/2} \][/tex]
Which can be simplified to:
[tex]\[ \frac{du}{dx} = \frac{3}{2\sqrt{x}} \][/tex]
3. Apply the chain rule:
The chain rule in Leibniz's notation states that:
[tex]\[ \frac{dy}{dx} = \frac{dy}{du} \cdot \frac{du}{dx} \][/tex]
Substitute the expressions we found for [tex]\(\frac{dy}{du}\)[/tex] and [tex]\(\frac{du}{dx}\)[/tex] into this formula:
[tex]\[ \frac{dy}{dx} = (20u^3) \cdot \left(\frac{3}{2\sqrt{x}}\right) \][/tex]
4. Substitute [tex]\( u = 3\sqrt{x} \)[/tex] into the expression:
Since [tex]\( u = 3\sqrt{x} \)[/tex], we can substitute this back into our expression for [tex]\( \frac{dy}{dx} \)[/tex]:
[tex]\[ u = 3\sqrt{x} \implies u^3 = (3\sqrt{x})^3 = 27x^{3/2} \][/tex]
Hence:
[tex]\[ \frac{dy}{dx} = 20(27x^{3/2}) \cdot \left(\frac{3}{2\sqrt{x}}\right) \][/tex]
Simplify this expression by multiplying the coefficients and combining the terms:
[tex]\[ \frac{dy}{dx} = 20 \cdot 27x^{3/2} \cdot \frac{3}{2\sqrt{x}} \][/tex]
Combine the terms:
[tex]\[ \frac{dy}{dx} = \frac{20 \cdot 27 \cdot 3}{2} x^{(3/2) - (1/2)} \][/tex]
[tex]\[ \frac{dy}{dx} = \frac{1620}{2} x^{1} \][/tex]
[tex]\[ \frac{dy}{dx} = 810x \][/tex]
Thus, the derivative [tex]\(\frac{dy}{dx}\)[/tex] is:
[tex]\[ \boxed{810x} \][/tex]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your visit. Our platform is always here to offer accurate and reliable answers. Return anytime. Thank you for visiting Westonci.ca. Stay informed by coming back for more detailed answers.
