Explore a vast range of topics and get informed answers at IDNLearn.com. Our platform is designed to provide accurate and comprehensive answers to any questions you may have.
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 your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Discover the answers you need at IDNLearn.com. Thank you for visiting, and we hope to see you again for more solutions.
