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Calculus

Implicit Differentiation

Concept Quizzes

Implicit Differentiation - Polynomials
Implicit Differentiation - Rational functions
Implicit Differentiation - Radical Functions
Implicit Differentiation - Exponential and Logarithmic Functions
Implicit Differentiation - Trigonometric Functions
Implicit Differentiation - Inverse Trigonometric Functions
Implicit Differentiation Problem Solving
Implicit Differentiation - Related Rates

Challenge Quizzes

Implicit Differentiation: Level 2 Challenges
Implicit Differentiation: Level 3 Challenges

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 $(a, b)?$

\frac{13}{244}

\frac{21}{244}

\frac{15}{244}

\frac{23}{244}

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by Brilliant Staff

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

\displaystyle{ \frac{dy}{dx}=\frac{6x^{2}}{\sqrt{\tan^{-1}(4x^{3})}(1+16x^{6})}}

\displaystyle{ \frac{dy}{dx}=\frac{6x^{2}}{\sqrt{\tan^{-1}(4x^{3})}}}

\displaystyle{ \frac{dy}{dx}=\frac{6x^{2}}{\sqrt{\tan^{-1}(4x^{3})}(1+4x^{3})}}

\displaystyle{ \frac{dy}{dx}=\frac{2x^{2}}{\sqrt{\tan^{-1}(4x^{3})}}}

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by Brilliant Staff

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 $a + b + c$ ?

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by Brilliant Staff

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

\displaystyle{ \frac{y e^{xy}-2\cos (2x+7y)}{x e^{xy}-7 \cos (2x+7y)}}

\displaystyle{ \frac{2\sin (2x+7y)-y e^{xy}}{x e^{xy}-7 \cos (2x+7y)}}

\displaystyle{ \frac{y e^{xy}-2\sin (2x+7y)}{x e^{xy}-7 \cos (2x+7y)}}

\displaystyle{ \frac{2\cos (2x+7y)-y e^{xy}}{x e^{xy}-7 \cos (2x+7y)}}

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by Brilliant Staff

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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by Brilliant Staff