Question

How do you find the derivative of $y=\sqrt{1+2x}$ ?

Answer 1

Hint: Problems on finding the derivative of an equation can be done by differentiating the equation with basic rules of differentiation. We will use the power rule and the chain rule to differentiate the given equation. Further simplifying we get the final solution.

Complete step-by-step solution:
The given equation we have is
$y=\sqrt{1+2x}$
We can rewrite the above equation as
$\Rightarrow y={{\left( 1+2x \right)}^{\dfrac{1}{2}}}$
Now for differentiation we apply chain rule for the right-hand part. According to the chain rule of differentiation: $\dfrac{d}{dx}f\left( u\left( x \right) \right)=f'\left( u\left( x \right) \right)\cdot u'\left( x \right)$
Here, the functions we have assumed are $f\left( u\left( x \right) \right)={{\left( 1+2x \right)}^{\dfrac{1}{2}}}$ and $u\left( x \right)=1+2x$ .
Taking the main equation $y={{\left( 1+2x \right)}^{\dfrac{1}{2}}}$ and differentiating both the sides, we get
$\dfrac{dy}{dx}=\dfrac{d\left\{ {{\left( 1+2x \right)}^{\dfrac{1}{2}}} \right\}}{dx}$
Using also the power rule which states that \[\dfrac{d}{dx}\left[ {{x}^{n}} \right]=n\cdot {{x}^{n-1}}\] we rewrite the above expression
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1}{2}\left( 1+2x \right)\dfrac{d\left( 1+2x \right)}{dx}$
Further completing the differentiation, we get
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1}{2}{{\left( 1+2x \right)}^{\dfrac{1}{2}-1}}\cdot 2$
Simplifying the above equation
$\Rightarrow \dfrac{dy}{dx}={{\left( 1+2x \right)}^{-\dfrac{1}{2}}}$
We know that terms having negative indices can be written as the reciprocal of that term with positive power. So, the term having a power of $-\dfrac{1}{2}$ can be converted into its reciprocal having the power of $+\dfrac{1}{2}$ as shown below
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1}{{{\left( 1+2x \right)}^{\dfrac{1}{2}}}}$
Rewriting the term having power of $\dfrac{1}{2}$as root of that term we get
$\Rightarrow \dfrac{dy}{dx}=\dfrac{1}{\sqrt{\left( 1+2x \right)}}$
Therefore, we conclude that the derivative of the given equation $y=\sqrt{1+2x}$ is $\dfrac{1}{\sqrt{\left( 1+2x \right)}}$.

Note: While applying the chain rule of differentiation we must follow the steps properly and avoid step jumps. As, inaccuracy in the result can occur. The derivation can also be done by the formal way of differentiation i.e., differentiation using the limits. Though, that process requires the concept of limits and the solution can be lengthy. So, we must avoid differentiation by formal method unless it is required.