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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.

Problem Solving - Basic

         

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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