Answer to Question #270947 in Calculus for pakie

Question

Consider the interesting curve below which is described by the equation ( ( )) 12 2 cosh sinh y 1 x − = + Determine an expression for dy y' dx = (your expression will contain both x and y functions). Use your calculations to answer questions 1 to 4 below. 1. The derivative with regards to x of ( ) 1 2 sinh y − is 2 2. Using the chain rule the derivative of ( ( )) 1 2 cosh sinh y − with regards to x is 3 3. The derivative of 2 1 x + is 2 4. The simplified version of dy y' dx = in terms of x and y is

Answer 1

$$\cosh(\sinh^{-1}(y^2))=\sqrt{1+x^2}$$

1.

$$(\sinh^{-1}(y^2))'_x=\dfrac{1}{\sqrt{1+y^4}}(2y)y'$$

2.

$$(\cosh(\sinh^{-1}(y^2)))'_x$$

$$=\sinh(\sinh^{-1}(y^2))\cdot(\sinh^{-1}(y^2))'_x$$

$$=y^2(\dfrac{1}{\sqrt{1+y^4}}(2y)y')=\dfrac{2y^3}{\sqrt{1+y^4}}y'$$

3.

$$(\sqrt{1+x^2})'_x=\dfrac{2x}{2\sqrt{1+x^2}}=\dfrac{x}{\sqrt{1+x^2}}$$

4.

$$\dfrac{2y^3}{\sqrt{1+y^4}}y'=\dfrac{x}{\sqrt{1+x^2}}$$

$$y'=\dfrac{x\sqrt{1+y^4}}{2y^3\sqrt{1+x^2}}$$