Edin's question via email about implicit differentiation

In summary, the given equation is differentiated with respect to x, and the resulting equations are used to find the first and second derivatives at the point (0, 2). The first derivative is found to be 0, and the second derivative is also found to be 0.
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A curve has equation $\displaystyle \begin{align*} y^3 + y + x\,y^2 = 10 + 4\sin{(x)} \end{align*}$.

(a) Determine y' at the point (0, 2)

(b) Determine y'' at the point (0,2)

(a) Differentiate both sides of the equation with respect to x:

$\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \left[ y^3 + y + x\,y^2 \right] &= \frac{\mathrm{d}}{\mathrm{d}x} \left[ 10 + 4\sin{(x)} \right] \\ 3\,y^2\,\frac{\mathrm{d}y}{\mathrm{d}x} + \frac{\mathrm{d}y}{\mathrm{d}x} + y^2 + 2\,x\,y\,\frac{\mathrm{d}y}{\mathrm{d}x} &= 4\cos{(x)} \\ \left( 3\,y^2 + 1 + 2\,x\,y \right) \frac{\mathrm{d}y}{\mathrm{d}x} &= 4\cos{(x)} - y^2 \\ \frac{\mathrm{d}y}{\mathrm{d}x} &= \frac{4\cos{(x)} - y^2}{3\,y^2 + 1 + 2\,x\,y} \end{align*}$

so at (0, 2) we have

$\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} &= \frac{4\cos{(0)} - 2^2}{3 \cdot 2^2 + 1 + 2 \cdot 0 \cdot 2} \\ &= \frac{0}{13} \\ &= 0 \end{align*}$(b) Differentiate both sides of the resulting equation with respect to x again...

$\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \left[ \left( 3\,y^2 + 1 + 2\,x\,y \right) \frac{\mathrm{d}y}{\mathrm{d}x} \right] &= \frac{\mathrm{d}}{\mathrm{d}x} \left[ 4\cos{(x)} - y^2 \right] \\ \left( 3\,y^2 + 1 + 2\,x\,y \right) \frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left( 6\,y\,\frac{\mathrm{d}y}{\mathrm{d}x} + 2\,y + 2\,x\,\frac{\mathrm{d}y}{\mathrm{d}x} \right) \frac{\mathrm{d}y}{\mathrm{d}x} &= -4\sin{(x)} - 2\,y\,\frac{\mathrm{d}y}{\mathrm{d}x} \end{align*}$

so at (0, 2) where $\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} = 0 \end{align*}$ we have

$\displaystyle \begin{align*} \left( 3\cdot 2^2 + 1 + 2\cdot 0 \cdot 2 \right) \frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left( 6 \cdot 2 \cdot 0 + 2\cdot 2 + 2 \cdot 0 \cdot 0 \right) \cdot 0 &= -4\sin{(0)} - 2\cdot 2 \cdot 0 \\ \left( 12 + 1 + 0 \right) \frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 0 &= 0 + 0 \\ 13\,\frac{\mathrm{d}^2y}{\mathrm{d}x^2} &= 0 \\ \frac{\mathrm{d}^2y}{\mathrm{d}x^2} &= 0 \end{align*}$
 
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(a) Differentiate both sides of the equation with respect to x:

$\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \left[ y^3 + y + x\,y^2 \right] &= \frac{\mathrm{d}}{\mathrm{d}x} \left[ 10 + 4\sin{(x)} \right] \\ 3\,y^2\,\frac{\mathrm{d}y}{\mathrm{d}x} + \frac{\mathrm{d}y}{\mathrm{d}x} + y^2 + 2\,x\,y\,\frac{\mathrm{d}y}{\mathrm{d}x} &= 4\cos{(x)} \\ \left( 3\,y^2 + 1 + 2\,x\,y \right) \frac{\mathrm{d}y}{\mathrm{d}x} &= 4\cos{(x)} - y^2 \\ \frac{\mathrm{d}y}{\mathrm{d}x} &= \frac{4\cos{(x)} - y^2}{3\,y^2 + 1 + 2\,x\,y} \end{align*}$

so at (0, 2) we have

$\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} &= \frac{4\cos{(0)} - 2^2}{3 \cdot 2^2 + 1 + 2 \cdot 0 \cdot 2} \\ &= \frac{0}{13} \\ &= 0 \end{align*}$(b) Differentiate both sides of the resulting equation with respect to x again...

$\displaystyle \begin{align*} \frac{\mathrm{d}}{\mathrm{d}x} \left[ \left( 3\,y^2 + 1 + 2\,x\,y \right) \frac{\mathrm{d}y}{\mathrm{d}x} \right] &= \frac{\mathrm{d}}{\mathrm{d}x} \left[ 4\cos{(x)} - y^2 \right] \\ \left( 3\,y^2 + 1 + 2\,x\,y \right) \frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left( 6\,y\,\frac{\mathrm{d}y}{\mathrm{d}x} + 2\,y + 2\,x\,\frac{\mathrm{d}y}{\mathrm{d}x} \right) \frac{\mathrm{d}y}{\mathrm{d}x} &= -4\sin{(x)} - 2\,y\,\frac{\mathrm{d}y}{\mathrm{d}x} \end{align*}$

so at (0, 2) where $\displaystyle \begin{align*} \frac{\mathrm{d}y}{\mathrm{d}x} = 0 \end{align*}$ we have

$\displaystyle \begin{align*} \left( 3\cdot 2^2 + 1 + 2\cdot 0 \cdot 2 \right) \frac{\mathrm{d}^2y}{\mathrm{d}x^2} + \left( 6 \cdot 2 \cdot 0 + 2\cdot 2 + 2 \cdot 0 \cdot 0 \right) \cdot 0 &= -4\sin{(0)} - 2\cdot 2 \cdot 0 \\ \left( 12 + 1 + 0 \right) \frac{\mathrm{d}^2y}{\mathrm{d}x^2} + 0 &= 0 + 0 \\ 13\,\frac{\mathrm{d}^2y}{\mathrm{d}x^2} &= 0 \\ \frac{\mathrm{d}^2y}{\mathrm{d}x^2} &= 0 \end{align*}$
Correct! Implicit differentiation was applied well.
 

FAQ: Edin's question via email about implicit differentiation

What is implicit differentiation?

Implicit differentiation is a method of finding the derivative of a function that is not expressed explicitly in terms of a single variable. It is used when the dependent variable cannot be isolated on one side of the equation.

How do you perform implicit differentiation?

To perform implicit differentiation, you follow the same rules of differentiation as you would for explicit functions, but you also use the chain rule for any variables that are not explicitly written in terms of the independent variable. You then solve for the derivative of the dependent variable.

Why is implicit differentiation useful?

Implicit differentiation allows us to find the derivative of functions that cannot be expressed explicitly in terms of a single variable. This is useful in many applications, such as finding the slope of a tangent line to a curve or finding the rate of change of a variable within a system of equations.

What are some examples of functions that require implicit differentiation?

Functions that involve multiple variables, trigonometric functions, logarithmic functions, and exponential functions often require implicit differentiation. For example, the equation x^2 + y^2 = 25 cannot be solved for y in terms of x, so implicit differentiation must be used to find the derivative of y.

What are the limitations of implicit differentiation?

Implicit differentiation can only be used to find the derivative of a function with respect to one variable. It cannot be used to find higher-order derivatives or to solve for the function itself. It also requires a good understanding of the chain rule, which can be challenging for some functions.

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