Answer :
Let's find the derivatives of the given functions step-by-step.
### Function 1: [tex]\( g(x) = 2x^3 - 7x^2 + 7 \)[/tex]
To find the derivative of [tex]\( g(x) \)[/tex], we will use the power rule, which states that [tex]\( \frac{d}{dx}[x^n] = nx^{n-1} \)[/tex].
1. Differentiate [tex]\( 2x^3 \)[/tex]:
[tex]\[ \frac{d}{dx}[2x^3] = 2 \cdot 3x^{3-1} = 6x^2 \][/tex]
2. Differentiate [tex]\( -7x^2 \)[/tex]:
[tex]\[ \frac{d}{dx}[-7x^2] = -7 \cdot 2x^{2-1} = -14x \][/tex]
3. The constant term [tex]\( 7 \)[/tex] has a derivative of:
[tex]\[ \frac{d}{dx}[7] = 0 \][/tex]
Combining these results, we get:
[tex]\[ g'(x) = 6x^2 - 14x \][/tex]
### Function 2: [tex]\( f(x) = (2x^3 - 7x^2 + 7)^5 \)[/tex]
To differentiate [tex]\( f(x) \)[/tex], we will use the chain rule. The chain rule states that if [tex]\( h(x) = (u(x))^n \)[/tex], then [tex]\( h'(x) = n(u(x))^{n-1} \cdot u'(x) \)[/tex], where [tex]\( u(x) \)[/tex] is the inner function.
1. Let [tex]\( u(x) = 2x^3 - 7x^2 + 7 \)[/tex]. We already found [tex]\( u'(x) \)[/tex] (which is the same as [tex]\( g'(x) \)[/tex]):
[tex]\[ u'(x) = 6x^2 - 14x \][/tex]
2. Function [tex]\( f(x) \)[/tex] can be written as [tex]\( f(x) = (u(x))^5 \)[/tex]. Applying the chain rule:
[tex]\[ f'(x) = 5(u(x))^4 \cdot u'(x) \][/tex]
Substituting [tex]\( u(x) \)[/tex] and its derivative [tex]\( u'(x) \)[/tex]:
[tex]\[ f'(x) = 5(2x^3 - 7x^2 + 7)^4 \cdot (6x^2 - 14x) \][/tex]
Therefore, the derivative of [tex]\( f(x) \)[/tex] is:
[tex]\[ f'(x) = (30x^2 - 70x)(2x^3 - 7x^2 + 7)^4 \][/tex]
### Summary of Derivatives:
- [tex]\( g'(x) = 6x^2 - 14x \)[/tex]
- [tex]\( f'(x) = (30x^2 - 70x)(2x^3 - 7x^2 + 7)^4 \)[/tex]
These are the derivatives of the functions [tex]\( g(x) \)[/tex] and [tex]\( f(x) \)[/tex] respectively.
### Function 1: [tex]\( g(x) = 2x^3 - 7x^2 + 7 \)[/tex]
To find the derivative of [tex]\( g(x) \)[/tex], we will use the power rule, which states that [tex]\( \frac{d}{dx}[x^n] = nx^{n-1} \)[/tex].
1. Differentiate [tex]\( 2x^3 \)[/tex]:
[tex]\[ \frac{d}{dx}[2x^3] = 2 \cdot 3x^{3-1} = 6x^2 \][/tex]
2. Differentiate [tex]\( -7x^2 \)[/tex]:
[tex]\[ \frac{d}{dx}[-7x^2] = -7 \cdot 2x^{2-1} = -14x \][/tex]
3. The constant term [tex]\( 7 \)[/tex] has a derivative of:
[tex]\[ \frac{d}{dx}[7] = 0 \][/tex]
Combining these results, we get:
[tex]\[ g'(x) = 6x^2 - 14x \][/tex]
### Function 2: [tex]\( f(x) = (2x^3 - 7x^2 + 7)^5 \)[/tex]
To differentiate [tex]\( f(x) \)[/tex], we will use the chain rule. The chain rule states that if [tex]\( h(x) = (u(x))^n \)[/tex], then [tex]\( h'(x) = n(u(x))^{n-1} \cdot u'(x) \)[/tex], where [tex]\( u(x) \)[/tex] is the inner function.
1. Let [tex]\( u(x) = 2x^3 - 7x^2 + 7 \)[/tex]. We already found [tex]\( u'(x) \)[/tex] (which is the same as [tex]\( g'(x) \)[/tex]):
[tex]\[ u'(x) = 6x^2 - 14x \][/tex]
2. Function [tex]\( f(x) \)[/tex] can be written as [tex]\( f(x) = (u(x))^5 \)[/tex]. Applying the chain rule:
[tex]\[ f'(x) = 5(u(x))^4 \cdot u'(x) \][/tex]
Substituting [tex]\( u(x) \)[/tex] and its derivative [tex]\( u'(x) \)[/tex]:
[tex]\[ f'(x) = 5(2x^3 - 7x^2 + 7)^4 \cdot (6x^2 - 14x) \][/tex]
Therefore, the derivative of [tex]\( f(x) \)[/tex] is:
[tex]\[ f'(x) = (30x^2 - 70x)(2x^3 - 7x^2 + 7)^4 \][/tex]
### Summary of Derivatives:
- [tex]\( g'(x) = 6x^2 - 14x \)[/tex]
- [tex]\( f'(x) = (30x^2 - 70x)(2x^3 - 7x^2 + 7)^4 \)[/tex]
These are the derivatives of the functions [tex]\( g(x) \)[/tex] and [tex]\( f(x) \)[/tex] respectively.
