How Do You Calculate These Mathematical Derivatives and Integrals?

In summary, differentiation is a mathematical process used to find the rate of change of a function and understand how it changes over time or in response to different inputs. To differentiate an equation, one must use rules such as the power rule, product rule, quotient rule, and chain rule. It has various applications in science, including physics, engineering, economics, and biology. It can also be used to solve real-world problems, but it does have limitations, such as only being applicable to continuous functions and not always yielding a solution.
  • #1
vorcil
398
0
1: Sin(4x^2) * 8x

2: Ln(3x) / x^2

3: If dy/dx = 2ex and y=6 when x=0, then y =

4: If ln(x^3) - ln(x) = 12, then x=e^(blank)

5: evaluate the integral by integrating it in terms of an area
(not sure how to use latex) integral from 1/2 -> 3/2 (2x-1)dx

my attempts =]

1: sin(4x^2)*8x
using the chainrule for the first bit of the equation
cos(4x^2)*8x
which is cos(4x^2)*8x*8
so 64x*cos(4x^2) is what i got

2: ln(3x) / x^2
using the quotient rule, lo*dhi - hi*dlo / lo^2
differentiating ln(3x) using chain rule i get 3(ln(3x)), x^2 = 2x
so quotient rule equation is
( (x^2)*3ln(3x) ) - (ln(3x) *2x) /(x^2)^2
simplifying
(x^2)(3ln(3x)) - ln(3x)*2x / x^8
(x^2)(2ln(3x))-2x / x^8
that's about as far as i got

3: If dy/dx = 2ex and y=6 when x=0, then y =

i tried to remember the old exponential differentiation rules,
if C=e^x then ln(c) = x
if c=ln(x) then e^x = x
not quite sure how to solve it from there :)


4: If ln(x^3) - ln(x) = 12, then x=e^(blank)
is this one of those exponential rules?

5: evaluate the integral by integrating it in terms of an area
(not sure how to use latex) integral from 1/2 -> 3/2 (2x-1)dx
 
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  • #2
For 1) use the product law

[tex]\frac{d}{dx}(uv)=v\frac{du}{dx}+u\frac{dv}{dx}[/tex]

2) Use the fact that d/dx(lnx)=1/x.

3) Integration is the reverse of differentiation. So d/dx(ex)=ex

4) Use your rules of logarithms here

5) Draw the line y=2x-1 and then draw x=1/2, then x=3/2. What figure do these lines and the x-axis form?
 
  • #3
thanks I've retried

1: sin(4x^2) * 8x
sin(4x^2) * 8 + cos(4x^2)*12x*8x
8sin(4x^2) + 96x*cos(4x^2)
is that right?

2: ln(3x)/x^2
(x^2 * 3/3x) - (ln(3x)*2x) / (x^2)^2
 
  • #4
3: If dy/dx = 2ex and y=6 when x=0, then y =

d/dx is also 2e^x + c = 6

2*e^x = 2, c=4
2e^x + 4 = 6
is this right?
 
  • #5
vorcil said:
thanks I've retried

1: sin(4x^2) * 8x
sin(4x^2) * 8 + cos(4x^2)*12x*8x
8sin(4x^2) + 96x*cos(4x^2)
is that right?

2: ln(3x)/x^2
(x^2 * 3/3x) - (ln(3x)*2x) / (x^2)^2

Where did you get the 12x from in the second line of 1?
2 looks correct

vorcil said:
3: If dy/dx = 2ex and y=6 when x=0, then y =

d/dx is also 2e^x + c = 6

2*e^x = 2, c=4
2e^x + 4 = 6
is this right?

ok if you know that when you differentiate ex with respect to x, you get ex+c.

and you differentiate y with respect to x to get ex, what do you get?
 
  • #6
rock.freak667 said:
Where did you get the 12x from in the second line of 1?
2 looks correct



ok if you know that when you differentiate ex with respect to x, you get ex+c.

and you differentiate y with respect to x to get ex, what do you get?

whoops should be 8x?
sin(4x^2)*8 + (cos(4x^2)*8x)*8x
8sin(4x^2)+(64x^2)*(cos(4x^2))
 
  • #7
5: evaluate the integral by integrating it in terms of an area
(not sure how to use latex) integral from 1/2 -> 3/2 (2x-1)dx

integral of 2x-1 = 2/2*x^2-x = (x^2)-x
((1.5^2)-1.5)-((0.5^2)-0.5) = 1

is that right?
 
  • #8
vorcil said:
5: evaluate the integral by integrating it in terms of an area
(not sure how to use latex) integral from 1/2 -> 3/2 (2x-1)dx

integral of 2x-1 = 2/2*x^2-x = (x^2)-x
((1.5^2)-1.5)-((0.5^2)-0.5) = 1

is that right?

I believe when they say in terms of an area, you should draw out what the integral represents. Draw y=2x-1,x=1/2,x=3/2 on the same graph and find the area of the enclosed figure.
 

FAQ: How Do You Calculate These Mathematical Derivatives and Integrals?

1. What is the purpose of differentiating an equation?

Differentiation is a mathematical process used to find the rate of change of a function with respect to one of its variables. It helps us understand how a function changes over time or in response to different inputs.

2. How do you differentiate an equation?

To differentiate an equation, you need to use the rules of differentiation, such as the power rule, product rule, quotient rule, and chain rule. These rules involve taking the derivative of each term in the equation and combining them to find the overall derivative.

3. What are the applications of differentiation in science?

Differentiation is widely used in various fields of science, including physics, engineering, economics, and biology. It helps us analyze and understand the behavior and properties of different natural phenomena, such as motion, growth, and change.

4. Can differentiation be used to solve real-world problems?

Yes, differentiation is a valuable tool for solving real-world problems. It can be used to optimize functions, find maximum and minimum values, and determine rates of change in various situations. It is also used in physics to calculate velocity and acceleration, and in economics to analyze supply and demand.

5. Are there any limitations to differentiation?

While differentiation is a powerful tool, it does have some limitations. It can only be applied to continuous functions, and some functions may not have a well-defined derivative. Additionally, differentiation may not always yield a solution to a problem, and in some cases, numerical methods may be needed to find an approximate solution.

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