find f(x), f(x), and f^(3)(x) for the following function. f(x)=2x^3 + 7x^2 + 4x f(x)=□ f(x)=□ f^(3)(x)=□
find f(x), f(x), and f^(3)(x) for the following function. f(x)=2x^3 + 7x^2 + 4x f(x)=□ f(x)=□ f^(3)(x)=□
Answer
Explanation:
Step1: Apply power - rule for first - derivative
The power - rule is $\frac{d}{dx}(x^n)=nx^{n - 1}$. For $f(x)=2x^{3}+7x^{2}+4x$, we have $f'(x)=\frac{d}{dx}(2x^{3})+\frac{d}{dx}(7x^{2})+\frac{d}{dx}(4x)$. So $f'(x)=2\times3x^{2}+7\times2x + 4=6x^{2}+14x + 4$.
Step2: Apply power - rule for second - derivative
Differentiate $f'(x)=6x^{2}+14x + 4$. Using the power - rule, $f''(x)=\frac{d}{dx}(6x^{2})+\frac{d}{dx}(14x)+\frac{d}{dx}(4)=6\times2x+14+0 = 12x+14$.
Step3: Apply power - rule for third - derivative
Differentiate $f''(x)=12x + 14$. Using the power - rule, $f^{(3)}(x)=\frac{d}{dx}(12x)+\frac{d}{dx}(14)=12+0 = 12$.
Answer:
$f'(x)=6x^{2}+14x + 4$ $f''(x)=12x + 14$ $f^{(3)}(x)=12$