Answer :
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]
