Initialize

Loading Student:-Calculus1

Solution #1

${\sinh }^{-1}\left(x^2\right)$$\stackrel{\text{differentiate w.r.t. x}}{\to }$$\frac{2x}{\sqrt{x^4\+1}}$$$

Solution #2

$\frac{\ⅆ}{\ⅆ x} {\sinh }^{-1}\left(x^2\right)$ = $\frac{2x}{\sqrt{x^4\+1}}$$$

Solution #3 (stepwise)

$$$\frac{\ⅆ}{\ⅆ x} {\sinh }^{-1}\left(x^2\right)$$\stackrel{\text{show solution steps}}{\to }$$\begin{array}{lll}& & \text{Differentiation Steps}\\ & & \frac{\ⅆ}{\ⅆx}\mathrm{arcsinh}\left(x^2\right)\\ \text{▫}& & \text{1. Apply the}\mathbf{\text{chain}}\text{rule to the term}\mathrm{arcsinh}\left(x^2\right)\\ & \text{◦}& \text{Recall the definition of the}\mathbf{\text{chain}}\text{rule}\\ & & \frac{\ⅆ}{\ⅆx}f\left(g\left(x\right)\right)=\mathrm{f\text{'}}\left(g\left(x\right)\right)\frac{\ⅆ}{\ⅆx}g\left(x\right)\\ & \text{◦}& \text{Outside function}\\ & & f\left(v\right)=\mathrm{arcsinh}\left(v\right)\\ & \text{◦}& \text{Inside function}\\ & & g\left(x\right)=x^2\\ & \text{◦}& \text{Derivative of outside function}\\ & & \frac{\ⅆ}{\ⅆv}f\left(v\right)=\frac{1}{\sqrt{v^2+1}}\\ & \text{◦}& \text{Apply composition}\\ & & \mathrm{f\text{'}}\left(g\left(x\right)\right)=\frac{1}{\sqrt{x^4+1}}\\ & \text{◦}& \text{Derivative of inside function}\\ & & \frac{\ⅆ}{\ⅆx}g\left(x\right)=2x\\ & \text{◦}& \text{Put it all together}\\ & & \frac{\ⅆ}{\ⅆx}f\left(g\left(x\right)\right)\frac{\ⅆ}{\ⅆx}g\left(x\right)=\frac{1}{\sqrt{x^4+1}}\cdot \left(2x\right)\\ & & \text{This gives:}\\ & & \frac{2x}{\sqrt{x^4+1}}\end{array}$$$

Figure 2.10.3(a)   Context Panel access to the Differentiation Rules