Solve Calculus Problems: Derivatives, Logs & More

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In summary, the derivative of the given functions with respect to x are: a) y = 5e^-4x ans: -20e ^-4x b) y = 1/2e^x^2 ans: xe^x^2 c) y = x^4 e^x ans: 4x^3 e^x + x^4 e^x d) y = e^-x (x) ans: - (x + 1)e^-x/(x^2) e) y = (1 + e^x)^1/2 ans: e^x/[2(1 + e^x)^1/2] f) y =
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what is the derivative of the following functions with respect to x:
a) y = 5e^-4x ans: -20e ^-4x
b) y = 1/2e^x^2 ans: xe^x^2
c)y = x^4 e^x ans: 4x^3 e^x + x^4 e^x
d) y = e^-x (x) ans: - (x + 1)e^-x/(x^2)
e) y = (1 + e^x)^1/2 ans: e^x/[2(1 + e^x)^1/2]
f) y = x + e ^ (x)^1/2 ans: 1 + [e ^(x)^1/2 / 2(x)^1/2]

3a) what is the equation of the tangent to the curve at the specified
point?
a) y = e^2x at (1, e^2) ans: y = 2e^2x - e^2
b) y = e^x^2/(x) at (1, e) ans: y = x

4)a) what is d^2/y/dx^2

a) y = x^2 e^-x ans: (x^2 -4x +2)e^-x
b) y = 4xe^x^2 ans: (16 x^3 + 24 x)e^x^2
c) y = e^-x sin x ans: -2e^-x cos x

5) how do you show that if y = e^x cos 2x then
d^2y/dx - 2(dy/dx) + 5y = 0?

=====
How do you evaluate log (subcript 2) 1/32?
b) log (g) 32 + log g(16) ?
c) log (2) 3
how do you solve for x? are there restrictions?
a) log (x) 25 = 2/3
b) log(7) [x + 7] + log(7) [x-7] = 0
 
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  • #2
Shouldn't this be in the Homework Help forum ?
 
  • #3
What are you stuck on? What can you do?
 
  • #4
Just a hint:

A function such as f(x) = e^x, you simply multiply the derivative of the exponent with the actual function. For example:

f(x) = e^(4x)

exponent is 4x and the derivative is 4 so:

f'(x) = 4 * e^(4x) = 4e^(4x)

That's the basic tenet of derivatives involving e.

If you need anymore help, just tell us where you are having trouble.
 

1. What is the first step in solving a calculus problem involving derivatives?

The first step in solving a calculus problem involving derivatives is to identify the function that needs to be differentiated. This function is usually denoted as f(x) or y.

2. How do I find the derivative of a logarithmic function?

To find the derivative of a logarithmic function, you can use the power rule for logarithms, which states that the derivative of log base a of x is 1 divided by x times the natural logarithm of a.

3. What is the chain rule in calculus and how is it applied?

The chain rule in calculus is used to find the derivative of a composite function, which is a function that is made up of two or more other functions. It states that the derivative of a composite function is equal to the derivative of the outer function multiplied by the derivative of the inner function.

4. How do I apply the product rule when solving a calculus problem?

The product rule in calculus is used to find the derivative of a function that is the product of two other functions. It states that the derivative of a product of two functions is equal to the first function times the derivative of the second function, plus the second function times the derivative of the first function.

5. What is the difference between a local maximum and a local minimum in calculus?

In calculus, a local maximum is the highest point on a graph within a certain interval, while a local minimum is the lowest point on a graph within a certain interval. These points are also known as critical points and can be found by setting the derivative of the function equal to zero and solving for x.

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