Chapter 2: Q 92 (page 212)

Use implicit differentiation and the power rule for integer powers (not the general power rule) to prove that

Short Answer

Expert verified

$y = x^{2 / 3} \Rightarrow y^{3} = x^{2} \Rightarrow \frac{d}{d x} y^{3} = \frac{d}{d x} x^{2} \Rightarrow 3 y^{2} \frac{d y}{d x} = 2 x \Rightarrow \frac{d y}{d x} = \frac{2 x}{3 y^{2}} \Rightarrow \frac{d y}{d x} = \frac{2}{3} x y^{- 2} \Rightarrow \Rightarrow \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x {(x^{2 / 3})}^{- 2} \Rightarrow \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x (x^{- 4 / 3}) \Rightarrow \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x^{1 - 4 / 3} \Rightarrow \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x^{- 1 / 3}$

Hence proved.

Step by step solution

Step 1. Given Information

We have given that :-

$\frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x^{- 1 / 3}$
We have to prove this derivative by using implicit differentiation and power rule for integer powers.

Step 2. Prove that ddx(x2/3)=23x-1/3

We have to find the derivative of $x^{2 / 3}$ .

Let :-

$y = x^{2 / 3}$ .

We can write it as :-

role="math" localid="1648652545614" $y^{3} = x^{2}$ .

Then by using implicit differentiation :-

role="math" localid="1648652570552" $\frac{d}{d x} y^{3} = \frac{d}{d x} x^{2}$

Then by using chain rule and power rule for integer powers, we have :-

role="math" localid="1648652595546" $3 y^{2} \frac{d y}{d x} = 2 x \Rightarrow \frac{d y}{d x} = \frac{2 x}{3 y^{2}} \Rightarrow \frac{d y}{d x} = \frac{2}{3} x y^{- 2}$

Put the value $y = x^{2 / 3}$ , then we have :-

role="math" localid="1648652616933" $\Rightarrow \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x {(x^{2 / 3})}^{- 2} \Rightarrow \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x (x^{- 4 / 3}) \Rightarrow \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x^{1 - 4 / 3} \Rightarrow \frac{d}{d x} (x^{2 / 3}) = \frac{2}{3} x^{- 1 / 3}$

This is the required value.

Hence proved.

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Use implicit differentiation and the power rule for integer powers (not the general power rule) to prove that

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Prove that ddx(x2/3)=23x-1/3

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