Sec 3.2: Product and Quotient Rules

Product Rule (formula derivation)

1. Find the derivatives of the following functions. You don't have to simplify your answer.  solution

(a) \(f(x)= (x^2+2x)(3x-5)\)        (b) \(f(x)= (5x-3x^5-x^7)(3x^2-5x)\) 

2. Find the derivatives of the following functions. You don't have to simplify your answer.  solution

(a) \(f(x)= \dfrac{x^2+2x}{5x-1}\)       (b) \(f(x)= \dfrac{7+3x^5-x^7}{3x^2-5x}\) 

3. Differentiate.   \( \displaystyle{f(x)=\frac{2-x e^{x}}{x+e^{x}}}\)    solution

4. If \( \displaystyle{f(x) = \frac{x^2+3x+1}{2x-5}} \),   find the value of   \( f'(0)\).     solution

5. If \(F(x)= (x^2+1)\, e^x \),   find \( F'(0) \),   and if \(\displaystyle{G(x)= \frac{x}{x-3}}\),   find \( G'(-1) \).    solution

6. If  \( g(x)= \sqrt{\frac{1}{x}} \),   find the value of  \( g'(4)\).  solution

7. Find an equation of the tangent line to the curve \(\displaystyle{ y=\frac{e^x}{1+x^2}}\) at the point \( (1, \frac{1}{2}e )\).   solution

Refer the table for the questions (8), (9), (10).  solution

\(x\)

\( f(x) \)

\( f'(x) \)

\( g(x) \)

\( g'(x) \)

0

  2

-2

4

1

1

2

3

1

2

(8)  If  \( H(x)=2 f(x)-g(x)\), find \( H'(0).\)  (9)  If  \( J(x)= f(x)\,g(x)\), find \( J'(1).\)   (10)  If  \( K(x)= e^x \,f(x) \), find \( K'(0).\)

 

11. Suppose that \(f(5)= 3, \, f^{\prime}(5)=2, \, g(5)=-6, \)   and  \(g^{\prime}(5)=7 \). Find the following values.    (a) \( (fg)'(5) \)   (b) \( (f/g)'(5) \)   (c) \( (g/f)'(5) \)    solution

12.  Find \( F'(\ln 2) \)  if  \(F(x)=  x \,e^x \).    solution

13. If  \( f(x)=x\, e^x \),   find the value of \( f''(0).\)      solution