Hello, I really need help on these two homework problem. ><

In problem #1, I believe it is related to the exponential functions and their derivatives section...in Calculus II? So, surely there should be a ln in the solution as well?

I would like a step by step process and explanation on how to solve these functions. I cannot seem to understand how to go about them correctly..

I would greatly appreciate it.
Thank you.

1) Solve the equation:

75/(16e^0.2x) = 30

2)Differentiate the function:

h(x) = 8^tanx

e^.2x = 75/(30*16)

.2x ln e = ln (75/480)
but ln e = 1
.2 x = ln (75/480)
=======================
8^tan x * ln 8 * (1/sec^2 x)

Thank you so much.

Although for problem number 1, I realized that I inputted it incorrectly (having problems with my + key).

It is actually:
75/(1+6e^0.2x) = 30

Would I still solve it the same way but subtract 1 to the other side?

make that sec^2 x and not 1/sec^2 x

Hello! I'd be happy to help you with your homework problems. Let's start with problem #1.

To solve the equation 75/(16e^0.2x) = 30, we can follow these steps:

Step 1: Multiply both sides of the equation by (16e^0.2x) to eliminate the fraction:
(75/(16e^0.2x)) * (16e^0.2x) = 30 * (16e^0.2x)

Step 2: Simplify and solve for x:
75 = 480e^0.2x

Step 3: Divide both sides of the equation by 480 to isolate e^0.2x:
75/480 = e^0.2x

Step 4: Calculate the left side of the equation:
0.15625 = e^0.2x

Step 5: To "undo" the exponential function e^0.2x, we can take the natural logarithm (ln) of both sides of the equation. This will give us:
ln(0.15625) = ln(e^0.2x)

Step 6: Simplify the right side using the property of ln(e^x) = x:
ln(0.15625) = 0.2x

Step 7: Finally, divide both sides of the equation by 0.2 to solve for x:
x = ln(0.15625) / 0.2

So, the solution to the equation 75/(16e^0.2x) = 30 is x = ln(0.15625) / 0.2.

Now, let's move on to problem #2, where we need to differentiate the function h(x) = 8^tanx.

To differentiate this function, we can follow these steps:

Step 1: Start with the function h(x) = 8^tanx.

Step 2: Apply the chain rule, which states that if we have a composite function f(g(x)), its derivative is given by f'(g(x)) * g'(x).

Step 3: Identify the outer function f(u) = 8^u, and the inner function g(x) = tanx.

Step 4: Derive the outer function f'(u) = (ln 8) * (8^u).

Step 5: Derive the inner function g'(x) = sec^2x.

Step 6: Apply the chain rule and multiply the derivatives: h'(x) = f'(g(x)) * g'(x)

Step 7: Substitute the derivatives we found: h'(x) = (ln 8) * (8^tanx) * sec^2x.

So, the derivative of the function h(x) = 8^tanx is h'(x) = (ln 8) * (8^tanx) * sec^2x.

I hope this step-by-step explanation helps you understand the process of solving these problems. Let me know if you have any further questions!