We consider the functions $T_n(x)$ defined as the $n$th partial derivative of Lebesgue's singular function $L_a(x)$ with respect to $a$ at $a=\frac{1}{2}$. This sequence includes a multiple of the Takagi function as the case $n=1$. We show that $T_n$ is continuous but nowhere differentiable for each $n$, and determine the Hölder order of $T_n$. From this, we derive that the Hausdorff dimension of the graph of $T_n$ is one. Using a formula of Lomnicki and Ulam, we obtain an arithmetic expression for $T_n(x)$ using the binary expansion of $x$, and use this to find the sets of points where $T_2$ and $T_3$ take on their absolute maximum and minimum values. We show that these sets are topological Cantor sets. In addition, we characterize the sets of local maximum and minimum points of $T_2$ and $T_3$.