Animated Solution for Mathematics - Differentiation: Given $y=\frac{5x}{3\sqrt{(1-x{)}^2}}+{\cos }^2(2x+1)$; Find $\frac{dy}{dx}$.

The function provided is $y=\frac{5x}{3\sqrt{(1-x{)}^2}}+{\cos }^2(2x+1)$. At first glance, it appears to be a standard differentiation exercise, but it contains a critical trap regarding the square root of a square.

Recall the fundamental identity $\sqrt{a^2}=|a|$. Applying this to our function, we obtain:

The presence of the modulus $|1-x|$ indicates that the function behaves differently depending on whether $x<1$ or $x>1$. We must analyze these two regions separately.

In this region, $1-x$ is positive, so $|1-x|=1-x$. The function simplifies to:

Applying the Quotient Rule to the rational term where $u=5x$ and $v=3(1-x)$, we find $u^{\prime }=5$ and $v^{\prime }=-3$. The derivative is given by:

Expanding the numerator yields $15-15x+15x=15$. Thus, the derivative of the rational part is:

In this region, $1-x$ is negative, so $|1-x|=-(1-x)=x-1$. The function becomes:

Applying the Quotient Rule with $u=5x$ and $v=3(x-1)$, we have $u^{\prime }=5$ and $v^{\prime }=3$. The derivative is:

Now, we differentiate the term ${\cos }^2(2x+1)$ using the Chain Rule. Let $f(u)=u^2$ where $u=\cos (2x+1)$:

This simplifies to $-4\sin (2x+1)\cos (2x+1)$. Using the double-angle identity $2\sin \theta \cos \theta =\sin (2\theta )$, we simplify this to:

By combining the rational and trigonometric derivatives, we arrive at the final piecewise derivative:

This result demonstrates the importance of respecting the domain and handling absolute values with precision. Always pause when encountering $\sqrt{a^2}$ to ensure your differentiation accounts for the sign of the expression.