
What is the derivative of ?
Answer
414.3k+ views
Hint: In this question , we need to find the derivative of . Mathematically, derivative is nothing but a rate of change of function with respect to an independent variable given in the function. Differentiation is used to calculate the derivatives. In two ways we can find the derivative of . We can find the derivative by taking logs on both sides. Also there is a direct formula to find the derivative of .
Logarithmic rule used :
Derivative formula used :
1.
2.
Complete step-by-step solution:
Given ,
Let us consider ,
On taking log on both sides,
We get,
We know that
By using that logarithmic rule,
We get,
On differentiating with respect to ,
We get ,
⇒
By cross multiplying,
We get,
We have already considered ,
By substituting the value of ,
We get,
Thus we get the derivative of is
Final answer :
The derivative of is
Note: Derivative helps in solving the problems in calculus and in differential equations. The derivative of with respect to is represented as . There are two types of derivative namely first order derivative and second order derivative. A simple example for a derivative is the derivative of is . Derivative is also applicable in trigonometric functions.
Alternative solution :
We can also find the derivative of by using a derivative formula.
Formula used :
Given,
Here
By substituting the value of in the formula,
We get,
Thus we get the derivative of is
Logarithmic rule used :
Derivative formula used :
1.
2.
Complete step-by-step solution:
Given ,
Let us consider ,
On taking log on both sides,
We get,
We know that
By using that logarithmic rule,
We get,
On differentiating
We get ,
⇒
By cross multiplying,
We get,
We have already considered
By substituting the value of
We get,
Thus we get the derivative of
Final answer :
The derivative of
Note: Derivative helps in solving the problems in calculus and in differential equations. The derivative of
Alternative solution :
We can also find the derivative of
Formula used :
Given,
Here
By substituting the value of
We get,
Thus we get the derivative of
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