Derivatives using Rules: Power Rule,    Formula Derivation     More Problems

A. Find \(f'(x)\)  of each of the following functions.   solution

(a) \( f(x)=x^2 \)       (b)  \( f(x)=x^5 \)      (c) \( f(x)=x^{1.8} \)      (d) \( f(x)=x^{\pi} \)

B. Find \(f'(x)\)  of each of the following functions.   solution

(a) \( f(x)= \sqrt{x} \)       (b)  \( f(x)=\sqrt[5]x\)      (c) \( f(x)=x^{3/4}\)     

C. Find \(f'(x)\)  of each of the following functions.   solution

(a) \( f(x) = \frac{1}{x} \)       (b)  \( f(x) = \frac{1}{x^5} \)      (c) \( f(x) = \frac{5}{x^{1/3}}\)     

D. Find \(f'(x)\)  of each of the following functions.  solution

(a) \( f(x)=5x^2+\frac{3}{x^2}\)       (b)  \( f(x)=4\sqrt{x}-10x+7 \)      (c) \( f(x)=\frac{1}{x^{4/3}}-\frac{x^3}{5}+7\pi \)     

E.  Answer the following questions (1)--(11).  (1)--(11) Solution (.pdf file)

(1) If  \( f(x)= x^3-4x^2+5\pi \),  find  \( f'(-2) \).    (2) If \( f(x)= \sqrt{x} \),  find  \( f'(4) \).

(3) If  \( f(x)= \frac{1}{\sqrt[3]{x}} \),  find  \( f'(1) \).    (4) If  \( f(x)= x^4 + 2 e^x \),  find   \( f'(0) \).

(5) If  \( f(x)= x -\sqrt{x}\),  find  \( f'(1) \).   (6) If  \(  f(s) = \sqrt{s} (s-1) \),  find \(f'(s)\). 

(7) If  \(\displaystyle{ f(t) = \frac{t^2-3t+1}{\sqrt{t}}} \),  find \( f'(t) \).  solution

(8) Find an equation of the tangent line to the curve  \(y= x^4-3x^2+5 \)  at  \( x=1\).  

(9) The equation of motion of a particle is  \( s=t^4-2t^3+t^2-t \).  Find the velocity after 1 second.

(10) Find the accelaration after 1 second of problem 9.

(11) If  \( f(x)=e^x \),  then find \( \ln(f'(2))\).

F. Find the points on the curve \(y=2x^3+3x^2-12x+1\) where the tangent is horizontal.  solution

G. Find the first and second derivatives of the function.   solution

1. \(\displaystyle{f(x)= x^4-3x^3+7x} \)    2. \(\displaystyle{f(x)= \frac{1}{x^5}} \)     3. \(\displaystyle{f(x)= \sqrt[5]x}\)   

H.  Find the third derivative of the given function.   solution

1. \(\displaystyle{f(x)= 2x^5+ 8x^2-3x+5}\)     2. \(\displaystyle{f(x)= \frac{1}{x}}\)