Question

If $u$ and $v$ are differentiable functions of $x$ and if $y = u + v$ then show that $\dfrac{{dy}}{{dx}} = \dfrac{{du}}{{dx}} + \dfrac{{dv}}{{dx}}$.

Answer 1

A function is said to be differentiable if its derivative exists and here, we need to prove the function as mentioned in the question let us consider the given term of $y$ as$y = u + v$, then differentiate both sides with respect to $x$. By this we can prove the equation LHS = RHS.

Complete step by step answer:
As per the statement given $u$and $v$are differentiable functions of $x$ and
$y = u + v$
Where we need to prove the functions of $x$ terms
For this let us consider the given term
$y = u + v$
Differentiating both sides with respect to $x$ we get
$\dfrac{{dy}}{{dx}} = \dfrac{{du}}{{dx}} + \dfrac{{dv}}{{dx}}$
Where,
Differentiation of $y$ with respect to $x$ is $\dfrac{{dy}}{{dx}}$.
Differentiation of $u$ with respect to $x$ is $\dfrac{{du}}{{dx}}$.
Differentiation of $v$ with respect to $x$ is $\dfrac{{dv}}{{dx}}$.
Therefore, we can see that by considering the given $y$ term we have proved the derivative function with respect to $x$.

Additional Information:
Differentiation is the algebraic method of finding the derivative for a function at any point.The derivative is a concept that is at the root of calculus.There are two ways of introducing the concept of derivatives, the geometrical way (as the slope of a curve) and the physical way (as a rate of change).

Note: Most functions are differentiable, which means that a derivative exists at every point on the function. Some functions, however, are not completely differentiable. Hence to find the differentiation function either with respect to $x$ or with respect to $y$, consider the given value then differentiate with respect to the terms asked or if applicable with some formulas or rules, then apply those to prove it.