Answer
Verified
376.8k+ views
Hint: In the given question, we need to find the derivative of the given function that involves square root within it. We know that derivatives are the rate of change of one variable with respect to another variable. It is also the slope of the tangent.
Complete step-by-step solution:
According to the given question, we need to find the derivative of the function $y=\sqrt{x}+\dfrac{1}{\sqrt{x}}$. Derivatives are basically known as the rate of change of one variable with respect to another variable.
Now, we know that derivative of ${{x}^{n}}=n{{x}^{n-1}}$ . So, using this in the given question we will get the derivative of the function which is denoted by $y'\left( x \right)$ .
Therefore,
\[\begin{align}
& y=\sqrt{x}+\dfrac{1}{\sqrt{x}} \\
& \Rightarrow y'\left( x \right)=\dfrac{1}{2\sqrt{x}}-\dfrac{1}{2x\sqrt{x}} \\
& \Rightarrow y'\left( x \right)=\dfrac{\left( x-1 \right)}{2x\sqrt{x}} \\
\end{align}\]
Now, in this first term we have n as half and for second term we have n as negative of half and then simply applying the previously mentioned formula used for derivation of polynomials we got the complete derivative of the function.
Here we have to find the value of $\dfrac{dy}{dx}$ at x=1 so we put this value in $y'\left( x \right)=\dfrac{\left( x-1 \right)}{2x\sqrt{x}}$.
Hence,
$\Rightarrow y'\left( x \right)=\dfrac{\left( 1-1 \right)}{2x\sqrt{1}} = 0 $
Hence, the derivative of the given function is \[y'\left( x \right)=\dfrac{\left( x-1 \right)}{2\sqrt{x}}\] and its value at x=1 is 0. So the option D is the correct answer.
Note: In such a type of question, we must be aware of the definition and how to evaluate the derivative. Also, we need to memorize the results or we can say derivatives of all kinds of functions in order to quickly write the result and move further. Some different functions whose derivatives must be memorized are trigonometric, logarithmic, exponential, polynomial etc.
Complete step-by-step solution:
According to the given question, we need to find the derivative of the function $y=\sqrt{x}+\dfrac{1}{\sqrt{x}}$. Derivatives are basically known as the rate of change of one variable with respect to another variable.
Now, we know that derivative of ${{x}^{n}}=n{{x}^{n-1}}$ . So, using this in the given question we will get the derivative of the function which is denoted by $y'\left( x \right)$ .
Therefore,
\[\begin{align}
& y=\sqrt{x}+\dfrac{1}{\sqrt{x}} \\
& \Rightarrow y'\left( x \right)=\dfrac{1}{2\sqrt{x}}-\dfrac{1}{2x\sqrt{x}} \\
& \Rightarrow y'\left( x \right)=\dfrac{\left( x-1 \right)}{2x\sqrt{x}} \\
\end{align}\]
Now, in this first term we have n as half and for second term we have n as negative of half and then simply applying the previously mentioned formula used for derivation of polynomials we got the complete derivative of the function.
Here we have to find the value of $\dfrac{dy}{dx}$ at x=1 so we put this value in $y'\left( x \right)=\dfrac{\left( x-1 \right)}{2x\sqrt{x}}$.
Hence,
$\Rightarrow y'\left( x \right)=\dfrac{\left( 1-1 \right)}{2x\sqrt{1}} = 0 $
Hence, the derivative of the given function is \[y'\left( x \right)=\dfrac{\left( x-1 \right)}{2\sqrt{x}}\] and its value at x=1 is 0. So the option D is the correct answer.
Note: In such a type of question, we must be aware of the definition and how to evaluate the derivative. Also, we need to memorize the results or we can say derivatives of all kinds of functions in order to quickly write the result and move further. Some different functions whose derivatives must be memorized are trigonometric, logarithmic, exponential, polynomial etc.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
Mark and label the given geoinformation on the outline class 11 social science CBSE
When people say No pun intended what does that mea class 8 english CBSE
Name the states which share their boundary with Indias class 9 social science CBSE
Give an account of the Northern Plains of India class 9 social science CBSE
Change the following sentences into negative and interrogative class 10 english CBSE
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
In Indian rupees 1 trillion is equal to how many c class 8 maths CBSE
Which are the Top 10 Largest Countries of the World?
How do you graph the function fx 4x class 9 maths CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
Difference Between Plant Cell and Animal Cell
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Why is there a time difference of about 5 hours between class 10 social science CBSE