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Implicit Differentiation

Not all equations can be written like y = f(x), so taking the derivative can be tricky. Save the mess and do it directly with implicit differentiation.

Implicit Differentiation Problem Solving

         

Let \((a, b)\) be the intersection point of the curve \(x^3+y^3+x^2y+7=0\) and the line \(9x+y=0.\) What is the slope of the tangent line to the curve \(x^3+y^3+x^2y+7=0\) at the point \((a, b)?\)

If \(\displaystyle y=\sqrt{\tan^{-1}(4x^{3})},\) what is \(\displaystyle \frac{dy}{dx}?\)

The equation of the line tangent to \((x^2 + y^2)^4 = 16x^2y^2\) at \((x,y) = (-1,1)\) is \(ay = bx + c\), where \(a\), \(b\) and \(c\) are positive integers. If \( a = 1 \), what is the value of \(a + b + c\)?

Given \(\displaystyle e^{xy}=\sin (2x+7y),\) what is \(\displaystyle \frac{dy}{dx}?\)

Suppose \(f(x)\) is a differentiable function that satisfies \(f(1)=16.\) If the derivative of the function \(g(x)=x \sqrt{f(x)}\) at the point \(x=1\) is \(5,\) what is the value of \(f'(1)?\)

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