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Find $ dy/dx $ by implicit differentiation.

$ x^4 + x^2y^2 + y^3 = 5 $

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$$y^{\prime}=\frac{-4 x^{3}-2 x y^{2}}{2 x^{2} y+3 y^{2}}=-\frac{2 x\left(2 x^{2}+y^{2}\right)}{y\left(2 x^{2}+3 y\right)}$$

00:39

Frank Lin

Calculus 1 / AB

Chapter 3

Differentiation Rules

Section 5

Implicit Differentiation

Derivatives

Differentiation

Parker S.

June 4, 2018

in the problem you set it equal to zero, when its supposed to be equal to 5.

Oregon State University

Harvey Mudd College

Baylor University

University of Michigan - Ann Arbor

Lectures

04:40

In mathematics, a derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a moving object with respect to time is the object's velocity. The concept of a derivative developed as a way to measure the steepness of a curve; the concept was ultimately generalized and now "derivative" is often used to refer to the relationship between two variables, independent and dependent, and to various related notions, such as the differential.

44:57

In mathematics, a differentiation rule is a rule for computing the derivative of a function in one variable. Many differentiation rules can be expressed as a product rule.

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Find $ dy/dx $ by implicit…

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In this problem were given an equation and we're asked to use implicit differentiation to find derivative of y with respect to x, so we're going to take the rivation terms with rspect to extent. So, let's start from the first term, we have 4 x cube plus forty. Second or we're going to get a product, so you have 2 x times y parent plus x, squared times 2 y times. Since y is a function of x. We also have y d x, plus 3 y square. Again y is a function of x. We have dy dx again and that is equal to 0, since y is his constant. So let's group terms with ye on the left hand side. So we of x, 2 x, squared y plus 3 y square, that is equal to the negative 2 x of y spread plus 2 x squared and from this we can see that the rate of y with respect to x is equal to negative of 2 x Times y square plus 2 x, squared divided by 2 x, squared plus 3 y, so 2 x, squared y plus 3 y square.

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