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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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