Category Archives: Derivative

Solved Problems in Derivatives

Problem 2-10: Diffrentiating

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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

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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

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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