Chapter 2: Q 91 (page 212)

Use implicit differentiation and the fact that $\frac{d}{d x} (x^{3}) = 3 x^{2}$ and $\frac{d}{d x} (x^{5}) = 5 x^{4}$ to prove that role="math" localid="1648649713044" $\frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{- 2 / 5}$ .

Short Answer

Expert verified

$y = x^{3 / 5} \Rightarrow y^{5} = x^{3} \Rightarrow 5 y^{4} \frac{d y}{d x} = 3 x^{2} \Rightarrow \frac{d y}{d x} = \frac{3 x^{2}}{5 y^{4}} \Rightarrow \frac{d y}{d x} = \frac{3}{5} x^{2} y^{- 4} \Rightarrow \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{2} {(x^{3 / 5})}^{- 4} \Rightarrow \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{2} (x^{3 / 5 \times - 4}) \Rightarrow \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{2} (x^{- 12 / 5}) \Rightarrow \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{2 - 12 / 5} \Rightarrow \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{- 2 / 5}$

Hence proved.

Step by step solution

Step 1. Given Information

We have given that :-

$\frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{- 2 / 5}$ .

We have to prove this derivative by using implicit differentiation and the following derivatives :-

$\frac{d}{d x} (x^{3}) = 3 x^{2}$ and $\frac{d}{d x} (x^{5}) = 5 x^{4}$ .

Step 2. Prove that ddx(x3/5)=35x-2/5

We have to find the derivative of $x^{3 / 5}$ .

Let :-

$y = x^{3 / 5}$ .

We can write it as :-

localid="1648650421493" $y^{5} = x^{3}$ .

Then by using implicit differentiation, given values localid="1648650137597" $\frac{d}{d x} (x^{3}) = 3 x^{2}$ and $\frac{d}{d x} (x^{5}) = 5 x^{4}$ and chain rule, we have :-

localid="1648650447179" $5 y^{4} \frac{d y}{d x} = 3 x^{2} \Rightarrow \frac{d y}{d x} = \frac{3 x^{2}}{5 y^{4}} \Rightarrow \frac{d y}{d x} = \frac{3}{5} x^{2} y^{- 4}$

Put the value $y = x^{3 / 5}$ , then we have :-

localid="1648650866497" $\frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{2} {(x^{3 / 5})}^{- 4} \Rightarrow \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{2} (x^{3 / 5 \times - 4}) \Rightarrow \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{2} (x^{- 12 / 5}) \Rightarrow \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{2 - 12 / 5} \Rightarrow \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{- 2 / 5}$

This is the required value.

Hence proved.

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Use implicit differentiation and the fact that $\frac{d}{d x} (x^{3}) = 3 x^{2}$ and $\frac{d}{d x} (x^{5}) = 5 x^{4}$ to prove that role="math" localid="1648649713044" $\frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{- 2 / 5}$ .

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Prove that ddx(x3/5)=35x-2/5

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Use implicit differentiation and the fact that ddx(x3)=3x2and ddx(x5)=5x4to prove that role="math" localid="1648649713044" ddx(x3/5)=35x-2/5.

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Prove that ddx(x3/5)=35x-2/5

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Decision Maths

Pure Maths

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Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Use implicit differentiation and the fact that $\frac{d}{d x} (x^{3}) = 3 x^{2}$ and $\frac{d}{d x} (x^{5}) = 5 x^{4}$ to prove that role="math" localid="1648649713044" $\frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{- 2 / 5}$ .