Introduction: If a function $y(x)$ is the sum of two functions $u(x)$ and $v(x)$, then we can apply the sum rule to determine the derivative of $y(x)$.
Rule: Let $y(x) = u(x) + v(x)$. Then:
$$ y'(x) = u'(x) + v'(x).$$
Example: Consider the function $y(x) = 5x^2 + \ln(x)$. This function can be denoted as $y(x)=u(x) + v(x)$, with $u(x)=5x^2$ and $v(x) = \ln(x)$. The derivative of $y(x)$ can be determined as follows (one might consider Derivative of elementary functions and the example at Scalar product rule):
$$\begin{align}
u'(x) &= 5 \cdot 2x = 10x,\\
v'(x) &= \dfrac{1}{x},\\
y'(x) &= 10x + \dfrac{1}{x}.
\end{align}$$
Rule: Let $y(x) = u(x) + v(x)$. Then:
$$ y'(x) = u'(x) + v'(x).$$
Example: Consider the function $y(x) = 5x^2 + \ln(x)$. This function can be denoted as $y(x)=u(x) + v(x)$, with $u(x)=5x^2$ and $v(x) = \ln(x)$. The derivative of $y(x)$ can be determined as follows (one might consider Derivative of elementary functions and the example at Scalar product rule):
$$\begin{align}
u'(x) &= 5 \cdot 2x = 10x,\\
v'(x) &= \dfrac{1}{x},\\
y'(x) &= 10x + \dfrac{1}{x}.
\end{align}$$