really stuck on these two, steps would be greatly appreciated. gotta find the derivative thanks!
first one:
f(t)=2^(log5t)
the 5 is the base
second one:
y=(2(x^2) - 1)^5 /(√(x+1)
woops, for the second one you have to use logarithmic differentiation
Sure! I can help you find the derivatives for both of these functions. Let's start with the first one:
First Function: f(t) = 2^(log5t)
To find the derivative, you can start by using the chain rule, which states that if you have a composite function, you can find its derivative by taking the derivative of the outer function, multiplied by the derivative of the inner function.
Step 1: Identify the outer function and inner function:
The outer function is 2^x, and the inner function is log5t.
Step 2: Find the derivative of the outer function:
The derivative of 2^x with respect to x is (ln(2)) * (2^x).
Step 3: Find the derivative of the inner function:
The derivative of log5t with respect to t is 1 / (t * ln(5)).
Step 4: Apply the chain rule:
Multiply the derivative of the outer function with the derivative of the inner function.
So, the derivative of f(t) = 2^(log5t) will be:
f'(t) = (ln(2)) * (2^(log5t)) * (1 / (t * ln(5)))
Now, let's move on to the second function:
Second Function: y = (2(x^2) - 1)^5 / (√(x + 1))
To find the derivative, we can use the quotient rule. The quotient rule states that if you have a function in the form of f(x)/g(x), the derivative can be found as (g(x) * f'(x) - f(x) * g'(x)) / (g(x))^2.
Step 1: Identify the numerator and denominator:
The numerator is (2(x^2) - 1)^5, and the denominator is √(x + 1).
Step 2: Expand and simplify the numerator:
The numerator can be expanded as (2^5) * ((x^2)^5) - 1^5, which simplifies as 32 * (x^10) - 1.
Step 3: Find the derivative of the numerator and denominator:
- Derivative of the numerator: Take the derivative of 32 * (x^10) - 1, which gives you 320 * (x^9).
- Derivative of the denominator: Take the derivative of √(x + 1), which can be written as (x + 1)^(1/2). The derivative is (1/2) * (x + 1)^(-1/2).
Step 4: Apply the quotient rule:
Using the quotient rule, the derivative of y = (2(x^2) - 1)^5 / (√(x + 1)) will be:
y' = [(√(x + 1)) * (320 * (x^9))] - [(2(x^2) - 1)^5 * (1/2) * (x + 1)^(-1/2)] / (√(x + 1))^2
I hope these step-by-step explanations help you find the derivatives for the given functions! Let me know if you have any further questions.