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)
for the second one you have to use logarithmic differentiation
recall that (a^u)' = lna a^u u', so we have
f' = ln2 2^log5t (log5t)'
Now, log5t = lnt/ln5, so
f' = ln2 2^log5t * 1/ln5 * 1/t
f' = ln2/ln5 * 1/t 2^log5t
Now, ln2/ln5 = log5(2), so finally,
f' = (log5(2) / t) 2^log5t
y = u^5/v where
u = 2x^2-1
v = √(x+1)
y' = (5u^4 u' v - u^5 v')/v^2
= u^4 (5vu' - v')/v^2
= (2x^2-1)^4 (5(2x^2-1) - 1/(2√(x+1)))/(x+1)
You can massage this as you want; one form is
(2x^2-1) (38x^2+40x+1)
------------------------------
2(x+1)√(x+1)
oops. forgot an ^4 on the last line
Sure! I can help you find the derivatives for both of these functions. Let's start with the first one:
Step-by-step solution for the first function, f(t) = 2^(log5t):
Step 1: Apply the power rule of differentiation:
To find the derivative of f(t), we can use the power rule, which states that if y = x^n, then the derivative of y with respect to x is dy/dx = n * x^(n-1).
Step 2: Rewrite the function using exponent rules:
We know that 2 can be represented as 2^1. So, we can rewrite the function as:
f(t) = (2^1)^(log5t).
Step 3: Apply the chain rule:
Now, we can apply the chain rule, which states that if y = f(g(x)), then the derivative of y with respect to x is dy/dx = f'(g(x)) * g'(x).
Applying the chain rule in this case, let:
u = log5t, and
f(u) = 2^u.
The derivative of f(u) = 2^u with respect to u is f'(u) = ln(2) * 2^u.
The derivative of u = log5t with respect to t is du/dt = 1 / (ln(5) * t).
Step 4: Calculate the derivative of f(t):
Now, we can substitute these values into the chain rule formula:
df/dt = f'(u) * du/dt
= ln(2) * 2^(log5t) * (1 / (ln(5) * t))
= (ln(2) / ln(5)) * 2^(log5t) / t.
So, the derivative of f(t) = 2^(log5t) is given by df/dt = (ln(2) / ln(5)) * 2^(log5t) / t.
Moving on to the second function:
Step-by-step solution for the second function, y = (2(x^2) - 1)^5 / (√(x+1)):
To find the derivative of this function, we can use logarithmic differentiation. The steps for logarithmic differentiation are as follows:
Step 1: Take the natural logarithm of both sides of the equation:
ln(y) = ln((2(x^2) - 1)^5 / (√(x + 1))).
Step 2: Simplify the expression using logarithm properties:
ln(y) = 5 * ln(2(x^2) - 1) - ln(√(x + 1)).
Step 3: Differentiate both sides of the equation with respect to x:
Apply the chain rule and product rule to find the derivatives of the terms on the right side of the equation.
For the first term, 5 * ln(2(x^2) - 1), we have:
d/dx[5 * ln(2(x^2) - 1)] = 5 * (1 / (2(x^2) - 1)) * 2x.
For the second term, ln(√(x + 1)), we have:
d/dx[ln(√(x + 1))] = (1 / (√(x + 1))) * (1 / 2).
Step 4: Simplify the derivatives from Step 3:
The derivative of the right side of the equation with respect to x is given by:
[5 * (1 / (2(x^2) - 1)) * 2x] - [(1 / (√(x + 1))) * (1 / 2)].
Step 5: Multiply both sides by y:
Now, multiply both sides of the equation by y, which is the original function.
y * [5 * (1 / (2(x^2) - 1)) * 2x] - [(1 / (√(x + 1))) * (1 / 2)] = y * d/dx[ln(y)].
Step 6: Substitute y back into the equation:
Substitute the original function, y = (2(x^2) - 1)^5 / (√(x + 1)), back into the equation.
[(2(x^2) - 1)^5 / (√(x + 1))] * [5 * (1 / (2(x^2) - 1)) * 2x] - [(1 / (√(x + 1))) * (1 / 2)] = [(2(x^2) - 1)^5 / (√(x + 1))] * d/dx[ln((2(x^2) - 1)^5 / (√(x + 1)))].
Step 7: Simplify and solve for the derivative:
Now, simplify and solve for the derivative by dividing both sides by [(2(x^2) - 1)^5 / (√(x + 1))].
[5 * (1 / (2(x^2) - 1)) * 2x] - [(1 / (√(x + 1))) * (1 / 2)] = d/dx[ln((2(x^2) - 1)^5 / (√(x + 1)))].
So, the derivative of y = (2(x^2) - 1)^5 / (√(x + 1)) with respect to x is given by:
d/dx = [5 * (1 / (2(x^2) - 1)) * 2x] - [(1 / (√(x + 1))) * (1 / 2)].
I hope this helps! Let me know if you have any further questions.