Join IDNLearn.com and become part of a knowledge-sharing community that thrives on curiosity. Discover reliable and timely information on any topic from our network of knowledgeable professionals.
Sagot :
Sure! Let's solve the given expressions step by step.
### 2) [tex]\( h^2 + 3h^3 \)[/tex]
We need to find the derivative of the expression [tex]\( h^2 + 3h^3 \)[/tex] with respect to [tex]\( h \)[/tex].
1. Compute the derivative of [tex]\( h^2 \)[/tex]:
- The power rule states that [tex]\(\frac{d}{dh} h^n = n h^{n-1}\)[/tex].
- For [tex]\( h^2 \)[/tex], [tex]\( n = 2 \)[/tex]: [tex]\(\frac{d}{dh} h^2 = 2h\)[/tex].
2. Compute the derivative of [tex]\( 3h^3 \)[/tex]:
- For [tex]\( 3h^3 \)[/tex], [tex]\( n = 3 \)[/tex]: [tex]\(\frac{d}{dh} 3h^3 = 3 \times 3h^{3-1} = 9h^2\)[/tex].
3. Add the results together:
- [tex]\(\frac{d}{dh} (h^2 + 3h^3) = 2h + 9h^2\)[/tex].
So, the derivative is:
[tex]\[ 2h + 9h^2 \][/tex]
### 3) [tex]\( 6h - 12 \)[/tex]
We need to find the derivative of the expression [tex]\( 6h - 12 \)[/tex] with respect to [tex]\( h \)[/tex].
1. Compute the derivative of [tex]\( 6h \)[/tex]:
- For [tex]\( 6h \)[/tex], [tex]\( n = 1 \)[/tex]: [tex]\(\frac{d}{dh} 6h = 6\)[/tex].
2. Compute the derivative of [tex]\( -12 \)[/tex]:
- The derivative of a constant is [tex]\( 0 \)[/tex].
3. Add the results together:
- [tex]\(\frac{d}{dh} (6h - 12) = 6 + 0 = 6\)[/tex].
So, the derivative is:
[tex]\[ 6 \][/tex]
### 4) [tex]\( 3i^5 - 15i^4 - 9i \)[/tex]
We need to find the derivative of the expression [tex]\( 3i^5 - 15i^4 - 9i \)[/tex] with respect to [tex]\( i \)[/tex].
1. Compute the derivative of [tex]\( 3i^5 \)[/tex]:
- For [tex]\( 3i^5 \)[/tex], [tex]\( n = 5 \)[/tex]: [tex]\(\frac{d}{di} 3i^5 = 3 \times 5i^{5-1} = 15i^4\)[/tex].
2. Compute the derivative of [tex]\( -15i^4 \)[/tex]:
- For [tex]\( -15i^4 \)[/tex], [tex]\( n = 4 \)[/tex]: [tex]\(\frac{d}{di} -15i^4 = -15 \times 4i^{4-1} = -60i^3\)[/tex].
3. Compute the derivative of [tex]\( -9i \)[/tex]:
- For [tex]\( -9i \)[/tex], [tex]\( n = 1 \)[/tex]: [tex]\(\frac{d}{di} -9i = -9\)[/tex].
4. Add the results together:
- [tex]\(\frac{d}{di} (3i^5 - 15i^4 - 9i) = 15i^4 - 60i^3 - 9\)[/tex].
So, the derivative is:
[tex]\[ 15i^4 - 60i^3 - 9 \][/tex]
### 5) [tex]\( ghg^2h^2 + gh \)[/tex]
We need to find the derivative of the expression [tex]\( ghg^2h^2 + gh \)[/tex] with respect to [tex]\( g \)[/tex] and [tex]\( h \)[/tex].
First, factor the expression:
1. Notice that the common factor in both terms is [tex]\( gh \)[/tex]:
- [tex]\( gh(g^2h + 1) \)[/tex].
Next, find the derivative of the factored form [tex]\( gh(g^2h + 1) \)[/tex]:
2. Compute the derivative of [tex]\( g^2h + 1 \)[/tex] with respect to [tex]\( g \)[/tex]:
- [tex]\(\frac{d}{dg} g^2h = 2gh\)[/tex].
- [tex]\(\frac{d}{dg} 1 = 0\)[/tex].
Combine the results to get the derivative with respect to [tex]\( g \)[/tex]:
- [tex]\( 2gh \)[/tex].
3. Compute the derivative of [tex]\( gh \)[/tex] with respect to [tex]\( g \)[/tex]:
- [tex]\(\frac{d}{dg} gh = h\)[/tex].
Combine the two results:
- [tex]\( 2gh + h \)[/tex].
4. Compute the derivative of [tex]\( g^2h + 1 \)[/tex] with respect to [tex]\( h \)[/tex]:
- [tex]\(\frac{d}{dh} g^2h = g^2\)[/tex].
- [tex]\(\frac{d}{dh} 1 = 0\)[/tex].
Combine the results to get the derivative with respect to [tex]\( h \)[/tex]:
- [tex]\( g^2 \)[/tex].
5. Compute the derivative of [tex]\( gh \)[/tex] with respect to [tex]\( h \)[/tex]:
- [tex]\(\frac{d}{dh} gh = g\)[/tex].
Combine the two results:
- [tex]\( g^2 + g \)[/tex].
6. Combine all partial results:
- [tex]\( gh (2gh + h + g^2) \)[/tex].
Therefore, the final derivative is:
[tex]\[ g^2h^2 + 4gh + g^3 \][/tex]
### 2) [tex]\( h^2 + 3h^3 \)[/tex]
We need to find the derivative of the expression [tex]\( h^2 + 3h^3 \)[/tex] with respect to [tex]\( h \)[/tex].
1. Compute the derivative of [tex]\( h^2 \)[/tex]:
- The power rule states that [tex]\(\frac{d}{dh} h^n = n h^{n-1}\)[/tex].
- For [tex]\( h^2 \)[/tex], [tex]\( n = 2 \)[/tex]: [tex]\(\frac{d}{dh} h^2 = 2h\)[/tex].
2. Compute the derivative of [tex]\( 3h^3 \)[/tex]:
- For [tex]\( 3h^3 \)[/tex], [tex]\( n = 3 \)[/tex]: [tex]\(\frac{d}{dh} 3h^3 = 3 \times 3h^{3-1} = 9h^2\)[/tex].
3. Add the results together:
- [tex]\(\frac{d}{dh} (h^2 + 3h^3) = 2h + 9h^2\)[/tex].
So, the derivative is:
[tex]\[ 2h + 9h^2 \][/tex]
### 3) [tex]\( 6h - 12 \)[/tex]
We need to find the derivative of the expression [tex]\( 6h - 12 \)[/tex] with respect to [tex]\( h \)[/tex].
1. Compute the derivative of [tex]\( 6h \)[/tex]:
- For [tex]\( 6h \)[/tex], [tex]\( n = 1 \)[/tex]: [tex]\(\frac{d}{dh} 6h = 6\)[/tex].
2. Compute the derivative of [tex]\( -12 \)[/tex]:
- The derivative of a constant is [tex]\( 0 \)[/tex].
3. Add the results together:
- [tex]\(\frac{d}{dh} (6h - 12) = 6 + 0 = 6\)[/tex].
So, the derivative is:
[tex]\[ 6 \][/tex]
### 4) [tex]\( 3i^5 - 15i^4 - 9i \)[/tex]
We need to find the derivative of the expression [tex]\( 3i^5 - 15i^4 - 9i \)[/tex] with respect to [tex]\( i \)[/tex].
1. Compute the derivative of [tex]\( 3i^5 \)[/tex]:
- For [tex]\( 3i^5 \)[/tex], [tex]\( n = 5 \)[/tex]: [tex]\(\frac{d}{di} 3i^5 = 3 \times 5i^{5-1} = 15i^4\)[/tex].
2. Compute the derivative of [tex]\( -15i^4 \)[/tex]:
- For [tex]\( -15i^4 \)[/tex], [tex]\( n = 4 \)[/tex]: [tex]\(\frac{d}{di} -15i^4 = -15 \times 4i^{4-1} = -60i^3\)[/tex].
3. Compute the derivative of [tex]\( -9i \)[/tex]:
- For [tex]\( -9i \)[/tex], [tex]\( n = 1 \)[/tex]: [tex]\(\frac{d}{di} -9i = -9\)[/tex].
4. Add the results together:
- [tex]\(\frac{d}{di} (3i^5 - 15i^4 - 9i) = 15i^4 - 60i^3 - 9\)[/tex].
So, the derivative is:
[tex]\[ 15i^4 - 60i^3 - 9 \][/tex]
### 5) [tex]\( ghg^2h^2 + gh \)[/tex]
We need to find the derivative of the expression [tex]\( ghg^2h^2 + gh \)[/tex] with respect to [tex]\( g \)[/tex] and [tex]\( h \)[/tex].
First, factor the expression:
1. Notice that the common factor in both terms is [tex]\( gh \)[/tex]:
- [tex]\( gh(g^2h + 1) \)[/tex].
Next, find the derivative of the factored form [tex]\( gh(g^2h + 1) \)[/tex]:
2. Compute the derivative of [tex]\( g^2h + 1 \)[/tex] with respect to [tex]\( g \)[/tex]:
- [tex]\(\frac{d}{dg} g^2h = 2gh\)[/tex].
- [tex]\(\frac{d}{dg} 1 = 0\)[/tex].
Combine the results to get the derivative with respect to [tex]\( g \)[/tex]:
- [tex]\( 2gh \)[/tex].
3. Compute the derivative of [tex]\( gh \)[/tex] with respect to [tex]\( g \)[/tex]:
- [tex]\(\frac{d}{dg} gh = h\)[/tex].
Combine the two results:
- [tex]\( 2gh + h \)[/tex].
4. Compute the derivative of [tex]\( g^2h + 1 \)[/tex] with respect to [tex]\( h \)[/tex]:
- [tex]\(\frac{d}{dh} g^2h = g^2\)[/tex].
- [tex]\(\frac{d}{dh} 1 = 0\)[/tex].
Combine the results to get the derivative with respect to [tex]\( h \)[/tex]:
- [tex]\( g^2 \)[/tex].
5. Compute the derivative of [tex]\( gh \)[/tex] with respect to [tex]\( h \)[/tex]:
- [tex]\(\frac{d}{dh} gh = g\)[/tex].
Combine the two results:
- [tex]\( g^2 + g \)[/tex].
6. Combine all partial results:
- [tex]\( gh (2gh + h + g^2) \)[/tex].
Therefore, the final derivative is:
[tex]\[ g^2h^2 + 4gh + g^3 \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. IDNLearn.com has the solutions to your questions. Thanks for stopping by, and come back for more insightful information.
