If 7x3 + 3y3 + 4x2 + 6x = 100, then (dy/dx)(2, 4) is
Implicit Differentiation Explained
When an equation involves both \(x\) and \(y\) variables mixed together, and it's not easy to solve for \(y\) explicitly in terms of \(x\), we use a technique called implicit differentiation. This method allows us to find the derivative \(\frac{dy}{dx}\) by differentiating both sides of the equation with respect to \(x\), treating \(y\) as an implicit function of \(x\). Remember to apply the chain rule when differentiating terms involving \(y\).
The given equation is: \[ 7x^3 + 3y^3 + 4x^2 + 6x = 100 \]
Differentiating the Equation Step-by-Step
We will differentiate each term of the equation with respect to \(x\).
- Derivative of \(7x^3\): Using the power rule \(\frac{d}{dx}(ax^n) = nax^{n-1}\), we get: \[ \frac{d}{dx}(7x^3) = 7 \cdot 3x^{3-1} = 21x^2 \]
- Derivative of \(3y^3\): Here, we treat \(y\) as a function of \(x\). Using the chain rule, we differentiate \(3y^3\) with respect to \(y\) and then multiply by \(\frac{dy}{dx}\): \[ \frac{d}{dx}(3y^3) = 3 \cdot 3y^{3-1} \cdot \frac{dy}{dx} = 9y^2 \frac{dy}{dx} \]
- Derivative of \(4x^2\): Using the power rule: \[ \frac{d}{dx}(4x^2) = 4 \cdot 2x^{2-1} = 8x \]
- Derivative of \(6x\): Using the power rule: \[ \frac{d}{dx}(6x) = 6 \cdot 1x^{1-1} = 6x^0 = 6 \]
- Derivative of \(100\): The derivative of a constant is \(0\): \[ \frac{d}{dx}(100) = 0 \]
Combining these derivatives, the differentiated equation becomes: \[ 21x^2 + 9y^2 \frac{dy}{dx} + 8x + 6 = 0 \]
Deriving the Formula for \(\frac{dy}{dx}\)
Now, we need to rearrange the differentiated equation to solve for \(\frac{dy}{dx}\).
First, move all terms not containing \(\frac{dy}{dx}\) to the right side of the equation: \[ 9y^2 \frac{dy}{dx} = -21x^2 - 8x - 6 \]
Next, divide both sides by \(9y^2\) to isolate \(\frac{dy}{dx}\): \[ \frac{dy}{dx} = \frac{-21x^2 - 8x - 6}{9y^2} \]
Evaluating the Derivative at the Point \((2, 4)\)
The question asks us to find the value of \(\frac{dy}{dx}\) at the specific point \((x, y) = (2, 4)\). We will substitute \(x=2\) and \(y=4\) into the expression for \(\frac{dy}{dx}\).
Substitute \(x=2\) into the numerator: \[ -21(2)^2 - 8(2) - 6 \] \[ = -21(4) - 16 - 6 \] \[ = -84 - 16 - 6 \] \[ = -100 - 6 \] \[ = -106 \]
Substitute \(y=4\) into the denominator: \[ 9(4)^2 \] \[ = 9(16) \] \[ = 144 \]
Now, combine the numerator and the denominator to find the value of \(\frac{dy}{dx}\) at \((2, 4)\): \[ \left(\frac{dy}{dx}\right)_{(2, 4)} = \frac{-106}{144} \]
Finally, simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is \(2\): \[ \frac{-106 \div 2}{144 \div 2} = \frac{-53}{72} \]
Therefore, the value of \(\left(\frac{dy}{dx}\right)_{(2, 4)}\) is \(-\frac{53}{72}\).
Important Questions from Differentiability
What is the value of f'(x) at x = 4 from the following table of values?
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