Find the derivative of f(x) = 4x^3 - 5x^4
Understand the Problem
The question asks us to find the derivative of the function f(x) = 4x^3 - 5x^4. This involves applying the power rule of differentiation to each term in the function. The power rule states that the derivative of x^n is nx^(n-1). We apply this rule to both 4x^3 and -5x^4, and then combine the results.
Answer
$f'(x) = 12x^2 - 20x^3$
Answer for screen readers
$f'(x) = 12x^2 - 20x^3$
Steps to Solve
- Apply the power rule to the first term
The first term is $4x^3$. Applying the power rule, we multiply the coefficient (4) by the exponent (3) and reduce the exponent by 1.
$$ \frac{d}{dx}(4x^3) = 4 \cdot 3 \cdot x^{3-1} = 12x^2 $$
- Apply the power rule to the second term
The second term is $-5x^4$. Applying the power rule, we multiply the coefficient (-5) by the exponent (4) and reduce the exponent by 1.
$$ \frac{d}{dx}(-5x^4) = -5 \cdot 4 \cdot x^{4-1} = -20x^3 $$
- Combine the derivatives
Now we combine the derivatives of the two terms to find the derivative of the entire function.
$$ f'(x) = 12x^2 - 20x^3 $$
$f'(x) = 12x^2 - 20x^3$
More Information
The derivative $f'(x) = 12x^2 - 20x^3$ represents the slope of the tangent line to the curve of the function $f(x) = 4x^3 - 5x^4$ at any given point x.
Tips
A common mistake is to forget to multiply the coefficient by the original exponent. Another one is to not subtract 1 from the exponent after multiplying. Also, students sometimes make errors with negative signs.
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