y = 7/(3x^4)
Understand the Problem
The question presents the equation (y = \frac{7}{3x^4}). The task is likely to analyze or manipulate this equation, potentially finding the derivative, integral, or graphing this function.
Answer
$\frac{dy}{dx} = -\frac{28}{3x^5}$
Answer for screen readers
$\frac{dy}{dx} = -\frac{28}{3x^5}$
Steps to Solve
- Rewrite the function
We can rewrite the function $y = \frac{7}{3x^4}$ using negative exponents to make differentiation easier. $$y = \frac{7}{3}x^{-4}$$
- Differentiate the function
Differentiate $y$ with respect to $x$ using the power rule: $\frac{d}{dx}(x^n) = nx^{n-1}$. $$\frac{dy}{dx} = \frac{7}{3} \cdot (-4)x^{-4-1}$$
- Simplify the derivative
Simplify the expression by multiplying the constants and combining the exponents. $$\frac{dy}{dx} = -\frac{28}{3}x^{-5}$$
- Rewrite with positive exponents
Rewrite the derivative using positive exponents. $$\frac{dy}{dx} = -\frac{28}{3x^5}$$
$\frac{dy}{dx} = -\frac{28}{3x^5}$
More Information
The derivative of the function $y = \frac{7}{3x^4}$ is $\frac{dy}{dx} = -\frac{28}{3x^5}$. This represents the instantaneous rate of change of $y$ with respect to $x$ at any given point.
Tips
A common mistake is forgetting to multiply the constant $\frac{7}{3}$ by the new coefficient after applying the power rule. Another common mistake is making an error when subtracting 1 from the exponent, especially when dealing with negative numbers. Make sure to rewrite negative exponents as positive exponents in the final answer as well.
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