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
To solve these questions, we'll break down the steps for each.
1. Simplify, and write in the base of 3: (9^x . 3^4x)^2
To simplify this expression, first, we need to apply the exponent rules. When you raise a power to another power, you multiply the exponents. Also, any number raised to the power of 2 gets squared.
Let's break it down:
(9^x . 3^4x)^2 = (3^2x . 3^4x)^2
Next, we use the property of exponentiation with the same base, which states that when you multiply bases with the same value, you add their exponents.
(3^2x . 3^4x)^2 = 3^(2x + 4x)^2
Simplify further:
3^(6x)^2 = 3^(12x)
Therefore, the simplified expression is 3^(12x).
2. Differentiate and simplify
a. y = 3x^4 - 4e^x + 8
To differentiate this function, we apply the power rule, which states that the derivative of x^n is n * x^(n-1). We'll differentiate each term separately.
The derivative of 3x^4 = 4 * 3 * x^(4-1) = 12x^3
The derivative of -4e^x = -4 * e^x (since the derivative of e^x is e^x)
The derivative of 8 = 0 (since 8 is a constant)
Putting it all together, the derivative of y = 3x^4 - 4e^x + 8 is 4(3x^3 - e^x).
b. y = (3 + 4e^x)^5
To differentiate this function, we'll need to use the chain rule. The chain rule states that the derivative of f(g(x)) is f'(g(x)) * g'(x). Let's break it down:
f(u) = u^5
g(x) = 3 + 4e^x
Applying the chain rule, we differentiate each part individually:
f'(u) = 5u^(5-1) = 5u^4
g'(x) = derivative of (3 + 4e^x) = 4e^x (since the derivative of e^x is e^x)
Now, we substitute these values back into the chain rule formula:
f'(g(x)) * g'(x) = 5(3 + 4e^x)^4 * 4e^x
Simplifying further, the derivative is 20e^x (3 + 4e^x)^4.
c. y = (e^x - 1) / x^2
To differentiate this function, we'll apply the quotient rule. The quotient rule states that the derivative of (f(x) / g(x)) is (f'(x) * g(x) - f(x) * g'(x)) / [g(x)]^2. Let's break it down:
f(x) = e^x - 1
g(x) = x^2
First, we'll calculate the derivatives of f(x) and g(x):
f'(x) = derivative of (e^x - 1) = e^x
g'(x) = derivative of (x^2) = 2x
Now, we substitute these values into the quotient rule formula:
[(e^x * x^2) - ((e^x - 1) * 2x)] / [x^2]^2
Simplifying further, we get (xe^x - 2e^x + 2) / x^3.
Therefore, the derivative of y = (e^x - 1) / x^2 is (xe^x - 2e^x + 2) / x^3.