Derivative from First Principles — Practice Problems and Worked Solutions

The formulas you need here

16 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Use the limit definition to find the derivative.

   $$f(x)=x^2,f^{\prime }(2)=\underset{h\to 0}{\lim }\frac{f(2+h)-f(2)}{h}$$

   Answer: $4$

   Step-by-step solution

   1. $$\frac{f(2+h)-f(2)}{h}=h+4$$
   2. $$\underset{h\to 0}{\lim }h+4=4$$
2. 2.

   Use the limit definition to find the derivative.

   $$f(x)=3x^2+x,f^{\prime }(1)=\underset{h\to 0}{\lim }\frac{f(1+h)-f(1)}{h}$$

   Answer: $7$

   Step-by-step solution

   1. $$\frac{f(1+h)-f(1)}{h}=3h+7$$
   2. $$\underset{h\to 0}{\lim }3h+7=7$$
3. 3.

   Use the limit definition to find the derivative.

   $$f(x)=x^3,f^{\prime }(-1)=\underset{h\to 0}{\lim }\frac{f(-1+h)-f(-1)}{h}$$

   Answer: $3$

   Step-by-step solution

   1. $$\frac{f(-1+h)-f(-1)}{h}=\frac{{\left(h-1\right)}^3+1}{h}$$
   2. $$\underset{h\to 0}{\lim }\frac{{\left(h-1\right)}^3+1}{h}=3$$
4. 4.

   Use the limit definition to find the derivative.

   $$f(x)=2x^3-x,f^{\prime }(2)=\underset{h\to 0}{\lim }\frac{f(2+h)-f(2)}{h}$$

   Answer: $23$

   Step-by-step solution

   1. $$\frac{f(2+h)-f(2)}{h}=2h^2+12h+23$$
   2. $$\underset{h\to 0}{\lim }2h^2+12h+23=23$$
5. 5.

   Use the limit definition to find the derivative.

   $$f(x)=x^2+4x-1,f^{\prime }(-2)=\underset{h\to 0}{\lim }\frac{f(-2+h)-f(-2)}{h}$$

   Answer: $0$

   Step-by-step solution

   1. $$\frac{f(-2+h)-f(-2)}{h}=h$$
   2. $$\underset{h\to 0}{\lim }h=0$$
6. 6.

   Use the limit definition to find the derivative.

   $$f(x)=x^4-2x^2,f^{\prime }(1)=\underset{h\to 0}{\lim }\frac{f(1+h)-f(1)}{h}$$

   Answer: $0$

   Step-by-step solution

   1. $$\frac{f(1+h)-f(1)}{h}=h\left(h^2+4h+4\right)$$
   2. $$\underset{h\to 0}{\lim }h\left(h^2+4h+4\right)=0$$
7. 7.

   Use the limit definition to find the derivative.

   $$f(x)=x^2,f^{\prime }(3)=\underset{h\to 0}{\lim }\frac{f(3+h)-f(3)}{h}$$

   Answer: $6$

   Step-by-step solution

   1. $$\frac{f(3+h)-f(3)}{h}=h+6$$
   2. $$\underset{h\to 0}{\lim }h+6=6$$
8. 8.

   Use the limit definition to find the derivative.

   $$f(x)=2x^2+5,f^{\prime }(1)=\underset{h\to 0}{\lim }\frac{f(1+h)-f(1)}{h}$$

   Answer: $4$

   Step-by-step solution

   1. $$\frac{f(1+h)-f(1)}{h}=2h+4$$
   2. $$\underset{h\to 0}{\lim }2h+4=4$$
9. 9.

   Use the limit definition to find the derivative.

   $$f(x)=x^2-3x,f^{\prime }(2)=\underset{h\to 0}{\lim }\frac{f(2+h)-f(2)}{h}$$

   Answer: $1$

   Step-by-step solution

   1. $$\frac{f(2+h)-f(2)}{h}=h+1$$
   2. $$\underset{h\to 0}{\lim }h+1=1$$
10. 10.

    Use the limit definition to find the derivative.

    $$f(x)=x^3,f^{\prime }(2)=\underset{h\to 0}{\lim }\frac{f(2+h)-f(2)}{h}$$

    Answer: $12$

    Step-by-step solution

    1. $$\frac{f(2+h)-f(2)}{h}=\frac{{\left(h+2\right)}^3-8}{h}$$
    2. $$\underset{h\to 0}{\lim }\frac{{\left(h+2\right)}^3-8}{h}=12$$
11. 11.

    Use the limit definition to find the derivative.

    $$f(x)=4x^2-x,f^{\prime }(-1)=\underset{h\to 0}{\lim }\frac{f(-1+h)-f(-1)}{h}$$

    Answer: $-9$

    Step-by-step solution

    1. $$\frac{f(-1+h)-f(-1)}{h}=4h-9$$
    2. $$\underset{h\to 0}{\lim }4h-9=-9$$
12. 12.

    Use the limit definition to find the derivative.

    $$f(x)=x^3+x,f^{\prime }(1)=\underset{h\to 0}{\lim }\frac{f(1+h)-f(1)}{h}$$

    Answer: $4$

    Step-by-step solution

    1. $$\frac{f(1+h)-f(1)}{h}=h^2+3h+4$$
    2. $$\underset{h\to 0}{\lim }{h}^2+3h+4=4$$
13. 13.

    Use the limit definition to find the derivative.

    $$f(x)=3x^2+2x,f^{\prime }(-2)=\underset{h\to 0}{\lim }\frac{f(-2+h)-f(-2)}{h}$$

    Answer: $-10$

    Step-by-step solution

    1. $$\frac{f(-2+h)-f(-2)}{h}=3h-10$$
    2. $$\underset{h\to 0}{\lim }3h-10=-10$$
14. 14.

    Use the limit definition to find the derivative.

    $$f(x)=x^3-4x,f^{\prime }(3)=\underset{h\to 0}{\lim }\frac{f(3+h)-f(3)}{h}$$

    Answer: $23$

    Step-by-step solution

    1. $$\frac{f(3+h)-f(3)}{h}=h^2+9h+23$$
    2. $$\underset{h\to 0}{\lim }{h}^2+9h+23=23$$
15. 15.

    Use the limit definition to find the derivative.

    $$f(x)=2x^3+x^2,f^{\prime }(1)=\underset{h\to 0}{\lim }\frac{f(1+h)-f(1)}{h}$$

    Answer: $8$

    Step-by-step solution

    1. $$\frac{f(1+h)-f(1)}{h}=2h^2+7h+8$$
    2. $$\underset{h\to 0}{\lim }2h^2+7h+8=8$$
16. 16.

    Use the limit definition to find the derivative.

    $$f(x)=x^4+x,f^{\prime }(-1)=\underset{h\to 0}{\lim }\frac{f(-1+h)-f(-1)}{h}$$

    Answer: $-3$

    Step-by-step solution

    1. $$\frac{f(-1+h)-f(-1)}{h}=\frac{h+{\left(h-1\right)}^4-1}{h}$$
    2. $$\underset{h\to 0}{\lim }\frac{h+{\left(h-1\right)}^4-1}{h}=-3$$

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