Implicit Differentiation

Implicit Differentiation Problem Solving


Let (a,b)(a, b) be the intersection point of the curve x3+y3+x2y+7=0x^3+y^3+x^2y+7=0 and the line 9x+y=0.9x+y=0. What is the slope of the tangent line to the curve x3+y3+x2y+7=0x^3+y^3+x^2y+7=0 at the point (a,b)?(a, b)?

If y=tan1(4x3),\displaystyle y=\sqrt{\tan^{-1}(4x^{3})}, what is dydx?\displaystyle \frac{dy}{dx}?

The equation of the line tangent to (x2+y2)4=16x2y2(x^2 + y^2)^4 = 16x^2y^2 at (x,y)=(1,1)(x,y) = (-1,1) is ay=bx+cay = bx + c, where aa, bb and cc are positive integers. If a=1 a = 1 , what is the value of a+b+ca + b + c?

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

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


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