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
y= 3x^4 - 4e^x + 8
dy/dx = 12 x^3 -4 e^x
= 4(3x^3 -e^x)
y = (3 + 4e^x)^5
Let u = 3+4e^x
y = u^5 dy/du = 5 y^4
dy/dx= dy/du*du/dx
dy/dx = 5(3 + 4e^x)^4*4e^x
= (20 e^x)(3 +4e^x)^4
You can do these. You won't learn without trying
To simplify and write in the base of 3: (9^x . 3^4x)^2, we can start by expanding the expression inside the parentheses.
First, we use the rule (a^b)^c = a^(b * c):
(9^x . 3^4x)^2 = (9^(x * 1) . 3^(4 * x))^2
= (9^(x) . 3^(4x))^2
Next, we use the rule (a * b)^c = a^c * b^c:
(9^(x) . 3^(4x))^2 = (9^x)^2 * (3^4x)^2
= 9^(x * 2) * 3^(4x * 2)
= 9^(2x) * 3^(8x)
Finally, we can write the expression in the base of 3 by rewriting powers of 3 using the rule a^b = c^(log_c(a^b)):
9^(2x) * 3^(8x) = 3^(2log_3(9x)) * 3^(8x)
= 3^(2 * log_3(9x) + 8x)
= 3^(2 * (log_3(9) + log_3(x)) + 8x)
= 3^(2 * (2 + log_3(x)) + 8x)
= 3^(4 + 2log_3(x) + 8x)
= 3^(2log_3(x) + 8x + 4)
= 3^(8x + 2log_3(x) + 4)
Therefore, the simplified expression written in the base of 3 is 3^(8x + 2log_3(x) + 4).
For the second question, to differentiate and simplify the given functions, we can use standard rules of differentiation:
a) y = 3x^4 - 4e^x + 8.
To differentiate this function, we apply the power rule and the exponential rule:
dy/dx = 4 * 3x^(4-1) - 4 * e^x + 0
= 12x^3 - 4e^x
Therefore, the derivative of y = 3x^4 - 4e^x + 8 is dy/dx = 12x^3 - 4e^x.
b) y = (3 + 4e^x)^5.
To differentiate this function, we use the chain rule and the power rule:
dy/dx = 5(3 + 4e^x)^4 * 4e^x
= 20e^x * (3 + 4e^x)^4
Therefore, the derivative of y = (3 + 4e^x)^5 is dy/dx = 20e^x * (3 + 4e^x)^4.
c) y = (e^x - 1) / x^2.
To differentiate this function, we use the quotient rule:
dy/dx = (x^2 * (e^x - 1)' - (e^x - 1) * (x^2)') / (x^2)^2
= (x^2 * e^x - x^2 * 1 - (e^x - 1) * 2x) / x^4
= (x^2 * e^x - x^2 - (2x * e^x - 2x)) / x^4
= (x^2 * e^x - 2x * e^x - x^2 + 2x) / x^4
= (x^2 * e^x - 2x * e^x - x^2 + 2x) / x^4
Therefore, the derivative of y = (e^x - 1) / x^2 is dy/dx = (x^2 * e^x - 2x * e^x - x^2 + 2x) / x^4.