Category Archives: Derivative

Solved Problems in Derivatives

Problem 2-10: Diffrentiating


Diffrentiate
a)  f_1\left(x\right)= \displaystyle\frac{1-2x^2}{1+x+x^2}
b)  f_2\left(x\right)= (1-2x^2)(1+x+x^2)

Solution
The following rules can be used:
I)  \displaystyle\frac{d}{dx}ax^n=anx^{n-1}
II)  \displaystyle\frac{d}{dx}\left(u+v\right)=\frac{du}{dx}+\frac{dv}{dx}

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Problem 2-9: Differentiating Polynomial and Rational Functions


Differentiate
a)  f\left(x\right)= 1+ 2x
b)  f\left(x\right)= 1 - 2x + 2x^2
c)  f\left(x\right)= 1- x^2 +\frac{1}{x}
d)  f\left(x\right)= 1 - x +\frac{1}{1-x}


Solution
The following rules can be used:
I)  \displaystyle\frac{d}{dx}ax^n=anx^{n-1}
II)  \displaystyle\frac{d}{dx}\left(u+v\right)=\frac{du}{dx}+\frac{dv}{dx}
III)  \displaystyle\frac{d}{dx}\frac{u}{v}=\frac{1}{v^2}\left(v\frac{du}{dx}-u\frac{dv}{dx}\right)
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Problem 2-2: Evaluating Derivative of Functions and the Tangent Lines


Find the derivative of  f(x) and the equation of the tangent line at  x_0=-1.

a)  f(x)=x^2
b)  f(x)=x^3+x+1
c)  f(x)=\frac{1}{x}

Solution
The equation of the tangent line at  {x}_{0} is  y = f'(x_0) (x-x_0) + f(x_0) .
a)  f(x)=x^2, f'(x)=2x
 y = f'(x_0) (x-x_0) + f(x_0) = 2x_0 (x-x_0)+x_0^2 = 2x_0 x-x_0^2
 y = -2x-1 .


b)  f(x)=x^3+x+1

 f'(x)=3x^2+1
 y=f'(x_0) (x-x_0) + f(x_0) =(3x_0^2+1) (x-x_0)+x_0^3+x_0+1=(3x_0^2+1)x-2x_0^3+1
 y=4x+3 .

c)  f(x)=\frac{1}{x}=x^{-1}
 f'(x)=-x^{-2}  f'(x)=-\frac{1}{x^2}
 y=f'(x_0) (x-x_0) + f(x_0) =-\frac{1}{x_0^2}(x-x_0)+\frac{1}{x_0}=-\frac{1}{x_0^2} x +\frac{2}{x_0}
 y = -x-2