a) To find the derivative of $y$ with respect to $x$ for the given function $y=\arctan (e^{4x})$, we can use the chain rule. Let's calculate it step by step.
Given: $y=\arctan (e^{4x})$
Taking the derivative of both sides with respect to $x$:
$\frac{d}{dx}(y)=\frac{d}{dx}(\arctan (e^{4x}))$
Applying the chain rule on the right-hand side:
$y^{\prime }=\frac{d}{du}(\arctan (u))·\frac{d}{dx}(e^{4x})$
The derivative of $\arctan (u)$ with respect to $u$ is $\frac{1}{1+u^2}$. In this case, $u=e^{4x}$, so:
$\frac{d}{du}(\arctan (u))=\frac{1}{1+(e^{4x}{)}^2}=\frac{1}{1+e^{8x}}$
The derivative of $e^{4x}$ with respect to $x$ is $4e^{4x}$. Substituting these values back into the equation:
$y^{\prime }=\frac{1}{1+e^{8x}}·\frac{d}{dx}(e^{4x})$
$y^{\prime }=\frac{1}{1+e^{8x}}·4e^{4x}$
Simplifying further:
$y^{\prime }=\frac{4e^{4x}}{1+e^{8x}}$
Therefore, the derivative of $y$ with respect to $x$ is $\frac{4e^{4x}}{1+e^{8x}}$.
b) To find the derivative of $y$ with respect to $x$ for the given function $y=\arcsin (f(x))$, we can also apply the chain rule. Let's solve it step by step.
Given: $y=\arcsin (f(x))$
Taking the derivative of both sides with respect to $x$:
$\frac{d}{dx}(y)=\frac{d}{dx}(\arcsin (f(x)))$
Applying the chain rule on the right-hand side:
$y^{\prime }=\frac{d}{du}(\arcsin (u))·\frac{d}{dx}(f(x))$
The derivative of $\arcsin (u)$ with respect to $u$ is $\frac{1}{\sqrt{1-u^2}}$. In this case, $u=f(x)$, so:
$\frac{d}{du}(\arcsin (u))=\frac{1}{\sqrt{1-(f(x){)}^2}}$
The derivative of $f(x)$ with respect to $x$ is $f^{\prime }(x)$. Substituting these values back into the equation:
$y^{\prime }=\frac{1}{\sqrt{1-(f(x){)}^2}}·\frac{d}{dx}(f(x))$
$y^{\prime }=\frac{1}{\sqrt{1-(f(x){)}^2}}·f^{\prime }(x)$
Therefore, the derivative of $y$ with respect to $x$ is $\frac{f^{\prime }(x)}{\sqrt{1-(f(x){)}^2}}$.