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Question: How do you find the derivative of \(\dfrac{1}{\left( x-5 \right)}\) ?...

How do you find the derivative of 1(x−5)\dfrac{1}{\left( x-5 \right)} ?

Explanation

Solution

Solving problems on differentiation can be easily done by using simple formulas of differentiation. We use the formula of power law for differentiation which is ddx[xn]=n⋅xn−1\dfrac{d}{dx}\left[ {{x}^{n}} \right]=n\cdot {{x}^{n-1}} . Further doing some simplifications we get the desired result.

Complete step-by-step solution:
The given expression we have is
1(x−5)\dfrac{1}{\left( x-5 \right)}
As the term x−5x-5 is in the denominator and only 11 is in the numerator the above expression can be written as x−5x-5 having a negative power. Hence, the above equation can be written as
⇒(x−5)−1....expression1\Rightarrow {{\left( x-5 \right)}^{-1}}....\text{expression1}
For differentiating the above expression, we will use one of the basic properties of differentiation which is the power rule of differentiation
According to the power rule ddx[xn]=n⋅xn−1\dfrac{d}{dx}\left[ {{x}^{n}} \right]=n\cdot {{x}^{n-1}}
Hence, taking the expression1\text{expression1} and differentiating it using the power rule we get
⇒ddx((x−5)−1)=(−1)(x−5)−1−1\Rightarrow \dfrac{d}{dx}\left( {{\left( x-5 \right)}^{-1}} \right)=\left( -1 \right){{\left( x-5 \right)}^{-1-1}}
Further simplifying the above expression, we can write as
⇒−(x−5)−2\Rightarrow -{{\left( x-5 \right)}^{-2}}
We know that numbers having negative indices can be written as the reciprocal of that number with positive indices.
Hence, the expression we got can be written as shown below
⇒−1(x−5)2\Rightarrow -\dfrac{1}{{{\left( x-5 \right)}^{2}}}
Therefore, we conclude to the result that the derivative of the expression 1(x−5)\dfrac{1}{\left( x-5 \right)} is −1(x−5)2-\dfrac{1}{{{\left( x-5 \right)}^{2}}}.

Note: 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. Also, while differentiating the expression using the power rule, we must be careful about the signs of the indices of the numbers, as they play an important role in differentiation.