Derivatives using Rules: Power Rule,    Formula Derivation     More Problems

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

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

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

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

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

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

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

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

  5. Find the derivative of the following functions.  soluion

    6. (a) \( f(r)=e^r+r^e \)   (b) \( f(x)=x(x^2-5)+e^5 \)

  6. Find the derivative of the following functions.  solution

    7. (a) \( f(s) = \sqrt{s} (s-1) \),     (b) \(f(x)=\dfrac{\sqrt{x}+x}{x^2} \)

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

  8. (a) If  \( f(x)= x^3-4x^2+5\pi \),  find  \( f'(-2) \), and (b) If \( f(x)= x^4 + 2 e^x \),  find   \( f'(0) \).  solution

  9. (a) If \( f(x)= \sqrt{x} \),  find  \( f'(4) \), and (b) if  \( g(x)= \frac{1}{\sqrt[3]{x}} \),  find  \( g'(1) \).    solution

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

  11. Find an equation of the tangent line to the curve  \(y= \sqrt[4]{x}-x \)  at the point  \((1,0)\).  solution

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

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

  14. For what value of \(x\) does the graph of \(f(x)=e^x-2x\) have a horizontal tangent line?  solution

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

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

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

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