Sure, let's go through this problem step-by-step to find the derivatives and their values at [tex]\( x = 3 \)[/tex].
Given:
[tex]\[ f(x) = 6x^3 - 6e^x \][/tex]
1. Finding the first derivative [tex]\( f'(x) \)[/tex]:
To find the first derivative, we differentiate [tex]\( f(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ f'(x) = \frac{d}{dx} (6x^3 - 6e^x) \][/tex]
Using the power rule and the chain rule:
[tex]\[ \frac{d}{dx} (6x^3) = 18x^2 \][/tex]
[tex]\[ \frac{d}{dx} (-6e^x) = -6e^x \][/tex]
Combining these results:
[tex]\[ f'(x) = 18x^2 - 6e^x \][/tex]
2. Finding [tex]\( f'(3) \)[/tex]:
Now, we substitute [tex]\( x = 3 \)[/tex] in the first derivative:
[tex]\[ f'(3) = 18(3^2) - 6e^3 \][/tex]
[tex]\[ f'(3) = 18(9) - 6e^3 \][/tex]
[tex]\[ f'(3) = 162 - 6e^3 \][/tex]
So, the value of the first derivative at [tex]\( x = 3 \)[/tex] is:
[tex]\[ f'(3) = 162 - 6e^3 \][/tex]
3. Finding the second derivative [tex]\( f''(x) \)[/tex]:
To find the second derivative, we differentiate [tex]\( f'(x) \)[/tex] with respect to [tex]\( x \)[/tex]:
[tex]\[ f''(x) = \frac{d}{dx} (18x^2 - 6e^x) \][/tex]
Using the power rule and the chain rule:
[tex]\[ \frac{d}{dx} (18x^2) = 36x \][/tex]
[tex]\[ \frac{d}{dx} (-6e^x) = -6e^x \][/tex]
Combining these results:
[tex]\[ f''(x) = 36x - 6e^x \][/tex]
4. Finding [tex]\( f''(3) \)[/tex]:
Now, we substitute [tex]\( x = 3 \)[/tex] in the second derivative:
[tex]\[ f''(3) = 36(3) - 6e^3 \][/tex]
[tex]\[ f''(3) = 108 - 6e^3 \][/tex]
So, the value of the second derivative at [tex]\( x = 3 \)[/tex] is:
[tex]\[ f''(3) = 108 - 6e^3 \][/tex]
To summarize, the solutions are:
[tex]\[
\begin{aligned}
f'(x) & = 18x^2 - 6e^x \\
f'(3) & = 162 - 6e^3
\end{aligned}
\][/tex]
[tex]\[
\begin{array}{l}
f''(x) = 36x - 6e^x \\
f''(3) = 108 - 6e^3
\end{array}
\][/tex]
