Question

1. Find the derivative of the given function, f(x)=1966

2. Find the derivative of the following function: f(x)=x²+6x+9

3. Find the slope of the tangent line to the graph of the following function at the indicated point, f(x)=3-2x at (-1, 5)

4. Find the derivative of the function: f(x)=x½

5. Find the derivative of the function f(x)=5x²(x+47)

6.) Find the derivative of the function: f(x)=5(x+47)²

7. Find the derivative of the function f(x)= 5x²/(x+47).

8. Which of the following functions are NOT differentiable?

a. f(x)=lxl

b. f(x)=(x+3)⁴

c. f(x)=mx+b

d. f(x)=1066

e. none of the above

9. Which of the following functions are NOT everywhere continous?

a. f(x)=x²-4/(x+2)

b. f(x)=(x+3)⁴

c. f(x)=1066

d. f(x)=mx+b

e. none of the above

Answer 1

Q.1:    Find the derivative of the given function, $f(x)=1966$

Solution

\begin{array}{l} f\left( x \right) = 1966\\ f'\left( x \right) = 0 \end{array}\               This is a constant function 

Q.2:    Find the derivative of the following function:$f(x)=x²+6x+9$

Solution

\begin{array}{l} f\left( x \right) = {x^2} + 6x + 9\\ f'\left( x \right) = 2x + 6 \end{array}\

Q.3:    Find the slope of the tangent line to the graph of the following function at the indicated point, $f(x)=3-2x$ at $(-1, 5)$

Solution

\begin{array}{l} f\left( x \right) = 3 - 2x\\ f'\left( x \right) = - 2 \end{array}\

The gradient of the tangent line is m = f'\left( x \right) = - 2\

Equation of tangent line is

\begin{array}{l} y - {y_1} = m\left( {x - {x_1}} \right)\\ y - 5 = - 2\left( {x + 1} \right)\\ 2x + y = 3 \end{array}\

Q.4:    Find the derivative of the function: $f(x)=x½$

Solutio

f\left( x \right) = {x^{{\textstyle{1 \over 2}}}}\

f'\left( x \right) = \frac{1}{2}{x^{{\textstyle{1 \over 2}} - 1}}\

f'\left( x \right) = \frac{1}{2}{x^{{\textstyle{{ - 1} \over 2}}}}\

Q.5:    Find the derivative of the function $f(x)=5x²(x+47)$

Solution

\begin{array}{l} f\left( x \right) = 5{x^2}\left( {x + 47} \right)\\ f\left( x \right) = 5{x^3} + 235{x^2}\\ f'\left( x \right) = 15{x^2} + 470x \end{array}\

Q.6:    Find the derivative of the function: $f(x)=5(x+47)²$

Solution

\begin{array}{l} f\left( x \right) = 5{\left( {x + 47} \right)^2}\\ f'\left( x \right) = 5 \cdot 2{\left( {x + 47} \right)^{2 - 1}} = 10\left( {x + 47} \right) \end{array}\

Q.7:    Find the derivative of the function $f(x)= \frac{5x²}{x+47}$

Solution

\begin{array}{l} f\left( x \right) = \frac{{5{x^2}}}{{\left( {x + 47} \right)}}\\ f'\left( x \right) = \frac{{\left( {x + 47} \right){{\left( {5{x^2}} \right)}^\prime } - \left( {5{x^2}} \right){{\left( {x + 47} \right)}^\prime }}}{{{{\left( {x + 47} \right)}^2}}}\\ f'\left( x \right) = \frac{{\left( {x + 47} \right)\left( {10x} \right) - \left( {5{x^2}} \right)\left( 1 \right)}}{{{{\left( {x + 47} \right)}^2}}}\\ f'\left( x \right) = \frac{{10{x^2} + 470x - 5{x^2}}}{{{{\left( {x + 47} \right)}^2}}}\\ f'\left( x \right) = \frac{{5{x^2} + 470x}}{{{{\left( {x + 47} \right)}^2}}} \end{array}\

Q.8: Which of the following functions are NOT differentiable?

a. f(x)=lxl

b. f(x)=(x+3)⁴

c. f(x)=mx+b

d. f(x)=1066

e. none of the above

Solution

f(x)=|x| is not differentiable at x = 0 

Q.9:    Which of the following functions are NOT everywhere continuous?

a. f(x)=x²-4/(x+2)

b. f(x)=(x+3)⁴

c. f(x)=1066

d. f(x)=mx+b

e. none of the above

Solution

The correct answer is (e)