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) \(
f(x)=\frac{1}{x^{4/3}}-\frac{x^3}{5}+7\pi \)
(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.
(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. \(\displaystyle{f(x)= \frac{1}{x}}\)