Answer :
The third derivative of [tex]\(f(x) = 3x^4 - 4x^3\) is \(f'''(x) = 72x - 24\).[/tex]
How to find the third derivative
The third derivative of[tex]\(f(x) = 3x^4 - 4x^3\)[/tex] can be found by taking successive derivatives.
First derivative:
[tex]\[f'(x) = \frac{d}{dx}(3x^4 - 4x^3) = 12x^3 - 12x^2.\][/tex]
Second derivative:
[tex]\[f''(x) = \frac{d}{dx}(12x^3 - 12x^2) = 36x^2 - 24x.\][/tex]
Third derivative:
[tex]\[f'''(x) = \frac{d}{dx}(36x^2 - 24x) = 72x - 24.\][/tex]
So, the third derivative of [tex]\(f(x) = 3x^4 - 4x^3\) is \(f'''(x) = 72x - 24\).[/tex]
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Final answer:
The third derivative of the function f(x) = 3x⁴ −4x³ is found by applying the power rule to each term. The third derivative is 72x - 24.
Explanation:
In order to find the third derivative of the function f(x), we will first take the first, second, and then third derivatives. Your function is f(x) = 3x⁴ −4x³.
First derivative f'(x) = 12x³ - 12x². You get this by multiplying the exponent by the coefficient for each term and then reducing the exponent by 1.
Proceed to take the second derivative f''(x) = 36x² - 24x, again applying the power rule.
For finding the third derivative f'''(x), apply the same rules. The third derivative of f(x) is f'''(x) = 72x - 24.
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