Answer :
Sure, let's find the derivative of the function step by step.
The given function is:
[tex]\[ f(x) = x^8 - 7x^4 + \pi^{30} + ex \][/tex]
We will differentiate each term of the function separately:
1. Derivative of [tex]\( x^8 \)[/tex]:
The derivative of [tex]\( x^8 \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( 8x^7 \)[/tex]. This follows from the power rule, which states that the derivative of [tex]\( x^n \)[/tex] is [tex]\( nx^{n-1} \)[/tex].
2. Derivative of [tex]\( -7x^4 \)[/tex]:
The derivative of [tex]\( -7x^4 \)[/tex] is [tex]\( -28x^3 \)[/tex]. Again, using the power rule ([tex]\(-7 \times 4 \times x^{4-1}\)[/tex]).
3. Derivative of [tex]\(\pi^{30}\)[/tex]:
[tex]\(\pi^{30}\)[/tex] is a constant, so its derivative is [tex]\(0\)[/tex].
4. Derivative of [tex]\( ex \)[/tex]:
The derivative of [tex]\( ex \)[/tex] is [tex]\( e \)[/tex]. This is because the derivative of a constant multiplied by [tex]\( x \)[/tex] is just the constant itself.
Combining all these results, the derivative of the function [tex]\( f(x) \)[/tex] is:
[tex]\[ f'(x) = 8x^7 - 28x^3 + 0 + e \][/tex]
Thus, the derivative of the function is:
[tex]\[ f'(x) = 8x^7 - 28x^3 + e \][/tex]
The given function is:
[tex]\[ f(x) = x^8 - 7x^4 + \pi^{30} + ex \][/tex]
We will differentiate each term of the function separately:
1. Derivative of [tex]\( x^8 \)[/tex]:
The derivative of [tex]\( x^8 \)[/tex] with respect to [tex]\( x \)[/tex] is [tex]\( 8x^7 \)[/tex]. This follows from the power rule, which states that the derivative of [tex]\( x^n \)[/tex] is [tex]\( nx^{n-1} \)[/tex].
2. Derivative of [tex]\( -7x^4 \)[/tex]:
The derivative of [tex]\( -7x^4 \)[/tex] is [tex]\( -28x^3 \)[/tex]. Again, using the power rule ([tex]\(-7 \times 4 \times x^{4-1}\)[/tex]).
3. Derivative of [tex]\(\pi^{30}\)[/tex]:
[tex]\(\pi^{30}\)[/tex] is a constant, so its derivative is [tex]\(0\)[/tex].
4. Derivative of [tex]\( ex \)[/tex]:
The derivative of [tex]\( ex \)[/tex] is [tex]\( e \)[/tex]. This is because the derivative of a constant multiplied by [tex]\( x \)[/tex] is just the constant itself.
Combining all these results, the derivative of the function [tex]\( f(x) \)[/tex] is:
[tex]\[ f'(x) = 8x^7 - 28x^3 + 0 + e \][/tex]
Thus, the derivative of the function is:
[tex]\[ f'(x) = 8x^7 - 28x^3 + e \][/tex]
