Q 91 Use implicit differentiation and... [FREE SOLUTION]

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

$y = x^{3 / 5} 鈬� y^{5} = x^{3} 鈬� 5 y^{4} \frac{d y}{d x} = 3 x^{2} 鈬� \frac{d y}{d x} = \frac{3 x^{2}}{5 y^{4}} 鈬� \frac{d y}{d x} = \frac{3}{5} x^{2} y^{- 4} 鈬� \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{2} {(x^{3 / 5})}^{- 4} 鈬� \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{2} (x^{3 / 5 脳 - 4}) 鈬� \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{2} (x^{- 12 / 5}) 鈬� \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{2 - 12 / 5} 鈬� \frac{d}{d x} (x^{3 / 5}) = \frac{3}{5} x^{- 2 / 5}$

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" $5 y^{4} \frac{d y}{d x} = 3 x^{2} 鈬� \frac{d y}{d x} = \frac{3 x^{2}}{5 y^{4}} 鈬� \frac{d y}{d x} = \frac{3}{5} x^{2} y^{- 4}$

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

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

This is the required value.

Hence proved.

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91影视

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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91影视

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

Pure Maths

Statistics

Logic and Functions

Discrete Mathematics

Theoretical and Mathematical Physics

Calculus

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