I have solved the following questions, let me know if they are correct, thanks.
1. Simplify, and write in the base of 3: (9^x . 3^4x)^2
answer: 3^12x
2. Differentiate and simplify
a. y= 3x^4 - 4e^x + 8
answer: 4(3x^3 - e^x)
b. y= (3 + 4e^x)^5
answer: 20e^x (3+ 4ex)^4
c. y= (e^x - 1) / x^2
answer: (xe^x - 2e^x +2) / x^3
All correct.
Sure, let's go through each question and determine if the answers are correct.
1. Simplify, and write in the base of 3: (9^x . 3^4x)^2
To simplify this expression, we can use the property of exponents that says (a^m)^n = a^(m*n). Applying this property, we get (9^x . 3^4x)^2 = 9^(2x) . (3^(4x))^2 = 9^(2x) * 3^(8x) = 3^(2x) * 3^(4x+8x) = 3^(2x) * 3^(12x) = 3^(2x+12x) = 3^(14x).
The correct answer is 3^(14x), not 3^(12x).
2. Differentiate and simplify
a. y = 3x^4 - 4e^x + 8
To differentiate this expression with respect to x, we can use the power rule for differentiation. The power rule states that d/dx (x^n) = n*x^(n-1).
Taking the derivative of each term separately, we get:
d/dx (3x^4) = 4 * 3 * x^(4-1) = 12x^3
d/dx (-4e^x) = -4 * e^x
d/dx (8) = 0 (the derivative of a constant is always zero)
So the simplified derivative is: 4(3x^3) - 4(e^x) + 0 = 12x^3 - 4e^x.
Your answer of 4(3x^3 - e^x) is correct.
b. y = (3 + 4e^x)^5
To differentiate this expression, we can use the chain rule. The chain rule states that if we have a function g(f(x)), then the derivative of g(f(x)) is g'(f(x)) * f'(x). In this case, our function g(u) is u^5, and our function f(x) is (3 + 4e^x).
So, applying the chain rule, we have:
d/dx (3 + 4e^x)^5 = 5(3 + 4e^x)^4 * d/dx (3 + 4e^x)
To find d/dx (3 + 4e^x), we differentiate each term separately. The derivative of a constant term (3) is zero, and the derivative of 4e^x is 4e^x.
Substituting back into the chain rule equation, we have:
5(3 + 4e^x)^4 * (0 + 4e^x) = 20e^x (3 + 4e^x)^4.
Your answer of 20e^x (3 + 4e^x)^4 is correct.
c. y = (e^x - 1) / x^2
To differentiate this expression, we can use the quotient rule. The quotient rule states that if we have a function f(x)/g(x), then the derivative is given by (g(x)*f'(x) - f(x)*g'(x)) / g(x)^2.
In this case, our numerator function f(x) is (e^x - 1), and our denominator function g(x) is x^2. Our numerator's derivative is e^x, and our denominator's derivative is 2x.
Applying the quotient rule, we have:
[(x^2)*(e^x) - (e^x - 1)*(2x)] / (x^2)^2 = (x^2e^x - 2xe^x + 2x) / x^4.
Your answer of (xe^x - 2e^x + 2) / x^3 is correct.
Congratulations on solving these questions correctly! If you have any more questions or need further explanations, feel free to ask.