Answer
Verified
481.2k+ views
Hint- Try to solve using the definition of derivative i.e. \[f\prime \left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h}\]
Let \[f\left( x \right) = x\]
We need to find the derivative of \[f\left( x \right)\]at\[{\text{x = 1}}\].
\[{\text{i}}{\text{.e}}{\text{. }}f\prime \left( 1 \right)\]
We know that,
\[f\prime \left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h}{\text{ }} \ldots \left( 1 \right)\]
Here, \[f\left( x \right) = x\]
So, \[{\text{ }}f\left( {x + h} \right) = x + h\]
Putting these values in equation $\left( 1 \right)$, we get
\[
f\prime \left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{\left( {x + h} \right) - x}}{h} \\
= \mathop {\lim }\limits_{h \to 0} \dfrac{{x + h - x}}{h} \\
= \mathop {\lim }\limits_{h \to 0} {\text{ }}\dfrac{h}{h} \\
= \mathop {\lim }\limits_{h \to 0} {\text{ }}1 \\
= 1 \\
\]
Hence, \[f\prime \left( x \right) = 1\]
Putting $x = 1$, we get
\[f\prime \left( 1 \right) = 1\]
Hence, the derivative of $x$ at $x=1$ is $1$.
Note- In order to calculate the derivative of a certain function, we assume that function to be $f\left( x \right)$ and apply the formula of the derivative and later put the value of $x$ for which the derivative is asked to calculate.
Let \[f\left( x \right) = x\]
We need to find the derivative of \[f\left( x \right)\]at\[{\text{x = 1}}\].
\[{\text{i}}{\text{.e}}{\text{. }}f\prime \left( 1 \right)\]
We know that,
\[f\prime \left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{f\left( {x + h} \right) - f\left( x \right)}}{h}{\text{ }} \ldots \left( 1 \right)\]
Here, \[f\left( x \right) = x\]
So, \[{\text{ }}f\left( {x + h} \right) = x + h\]
Putting these values in equation $\left( 1 \right)$, we get
\[
f\prime \left( x \right) = \mathop {\lim }\limits_{h \to 0} \dfrac{{\left( {x + h} \right) - x}}{h} \\
= \mathop {\lim }\limits_{h \to 0} \dfrac{{x + h - x}}{h} \\
= \mathop {\lim }\limits_{h \to 0} {\text{ }}\dfrac{h}{h} \\
= \mathop {\lim }\limits_{h \to 0} {\text{ }}1 \\
= 1 \\
\]
Hence, \[f\prime \left( x \right) = 1\]
Putting $x = 1$, we get
\[f\prime \left( 1 \right) = 1\]
Hence, the derivative of $x$ at $x=1$ is $1$.
Note- In order to calculate the derivative of a certain function, we assume that function to be $f\left( x \right)$ and apply the formula of the derivative and later put the value of $x$ for which the derivative is asked to calculate.
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
Give 10 examples for herbs , shrubs , climbers , creepers
Change the following sentences into negative and interrogative class 10 english CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you graph the function fx 4x class 9 maths CBSE
Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
Application to your principal for the character ce class 8 english CBSE
Write a letter to the principal requesting him to grant class 10 english CBSE