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May 2010
3
0
1. suppose that f(x) and g(x) are twise differentiable at x=a,
show that
(fg)"(a)= f(a) g"(a) + 2f'(a) g'(a) + f"(a) g(a)
_______________________________________________________________

2. Use the definition of derivative to show that for the function

f(x)=3(x)^(1/3)

(f of x is equal to 3 x to the power 1/3)

we have f'(a)=a^(2/3), for every "a" belongs to "R" (Set of real numbers)

[Use the identity (a^3)-(b^3)= (a-b)((a^2)+ab+(b^2))]
_______________________________________________________________

3. Differentiate the following and prove the necessary expressions as given

a) If y=xtanx

show that

x sin²x (dy/dx)=x²tan²x +y sin²x
_____
b) y= (x+√ x²-1 )ⁿ
prove that

(x²-1) (dy/dx)²=n²y²


Plz help me to solve these. (Crying)
 

Prove It

MHF Helper
Aug 2008
12,883
4,999
2. Use the definition of derivative to show that for the function

f(x)=3(x)^(1/3)

(f of x is equal to 3 x to the power 1/3)

we have f'(a)=a^(2/3), for every "a" belongs to "R" (Set of real numbers)

[Use the identity (a^3)-(b^3)= (a-b)((a^2)+ab+(b^2))]
_________________________________________________
\(\displaystyle f(x) = 3x^{\frac{1}{3}}\).


\(\displaystyle f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}\)

\(\displaystyle = \lim_{h \to 0}\frac{3(x + h)^{\frac{1}{3}} - 3x^{\frac{1}{3}}}{h}\)

\(\displaystyle = \lim_{h \to 0}\frac{3[(x + h)^{\frac{1}{3}} - x^{\frac{1}{3}}]}{h}\)

\(\displaystyle = \lim_{h \to 0}\frac{3\left[\frac{(x + h) - x}{(x + h)^{\frac{2}{3}} + + (x + h)^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}\right]}{h}\)

\(\displaystyle = \lim_{h \to 0}\frac{\frac{3h}{(x + h)^{\frac{2}{3}} + (x + h)^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}}{h}\)

\(\displaystyle = \lim_{h \to 0}\frac{3}{(x + h)^{\frac{2}{3}} + (x + h)^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}\)

\(\displaystyle = \frac{3}{x^{\frac{2}{3}} + x^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}\)

\(\displaystyle = \frac{3}{3x^{\frac{2}{3}}}\)

\(\displaystyle = \frac{1}{x^{\frac{2}{3}}}\)

\(\displaystyle = x^{-\frac{2}{3}}\)
 
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May 2010
3
0
\(\displaystyle f(x) = 3x^{\frac{1}{3}}\).


\(\displaystyle f'(x) = \lim_{h \to 0}\frac{f(x + h) - f(x)}{h}\)

\(\displaystyle = \lim_{h \to 0}\frac{3(x + h)^{\frac{1}{3}} - 3x^{\frac{1}{3}}}{h}\)

\(\displaystyle = \lim_{h \to 0}\frac{3[(x + h)^{\frac{1}{3}} - x^{\frac{1}{3}}]}{h}\)

\(\displaystyle = \lim_{h \to 0}\frac{3\left[\frac{(x + h) - x}{(x + h)^{\frac{2}{3}} + + (x + h)^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}\right]}{h}\)

\(\displaystyle = \lim_{h \to 0}\frac{\frac{3h}{(x + h)^{\frac{2}{3}} + (x + h)^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}}{h}\)

\(\displaystyle = \lim_{h \to 0}\frac{3}{(x + h)^{\frac{2}{3}} + (x + h)^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}\)

\(\displaystyle = \frac{3}{x^{\frac{2}{3}} + x^{\frac{1}{3}}x^{\frac{1}{3}} + x^{\frac{2}{3}}}\)

\(\displaystyle = \frac{3}{3x^{\frac{2}{3}}}\)

\(\displaystyle = \frac{1}{x^{\frac{2}{3}}}\)

\(\displaystyle = x^{-\frac{2}{3}}\)
Thanks(Rofl)
 

HallsofIvy

MHF Helper
Apr 2005
20,249
7,909
1. suppose that f(x) and g(x) are twise differentiable at x=a,
show that
(fg)"(a)= f(a) g"(a) + 2f'(a) g'(a) + f"(a) g(a)
Use the product rule- twice.

_______________________________________________________________

2. Use the definition of derivative to show that for the function

f(x)=3(x)^(1/3)

(f of x is equal to 3 x to the power 1/3)

we have f'(a)=a^(2/3), for every "a" belongs to "R" (Set of real numbers)

[Use the identity (a^3)-(b^3)= (a-b)((a^2)+ab+(b^2))]
_______________________________________________________________

3. Differentiate the following and prove the necessary expressions as given

a) If y=xtanx

show that

x sin²x (dy/dx)=x²tan²x +y sin²x
Okay, what did you get for dy/dx?
_____
b) y= (x+√ x²-1 )ⁿ
prove that

(x²-1) (dy/dx)²=n²y²
Again, what did you get for dy/dx?

Plz help me to solve these. (Crying)