Q10P - Mathematical Methods in Physical Sciences

Question

Verify each of the following by using equations (11.4), (12.2), and (12.3).

$\frac{\mathbf{d}}{\mathbf{dz}}\mathbf{cosh}\mathbf{ }\mathbf{z}\mathbf{=}\mathbf{sinhz}$

Step-by-Step Solution

Answer

The equation $\frac{\mathrm{d}}{\mathrm{dz}}\mathrm{coshz}=\sinh \mathrm{z}$ is verified using the equations (11.4), (12.2) and (12.3).

Step 1: Given information

The given function is $\frac{\mathrm{d}}{\mathrm{dz}}\mathrm{coshz}=\sinh \mathrm{z}$.

Step 2: Definition of Hyperbolic Function.

The term "Hyperbolic Function" refers to the relationship between a point on a hyperbola's distance from its origin and its coordinate axes, expressed as a function of an angle.

Step 3: Use exponential form to expand the equation

The exponential form of the given equation is,

$\cos h z=\frac{e^z{e}^{-z}}{2}$                                                                                                        …. (1)

 

Differentiate equation (1) with respect to z.

$\frac{d}{dz}\cos h z=\frac{1}{2}\times \frac{d}{dz}\left(e^z+e^{-z}\right)$                                                                                      …. (2)

 

Open equation (2) using algebra of derivatives.

$$\begin{gathered} \frac{d}{dz}\cos h z=\frac{1}{2}\times \left(e^z-e^{-z}\right) \\ \frac{d}{dz}\cos h z=\sin h z \end{gathered}$$

 

Hence the equation is verified.