Find the derivative.
11) f(t)=(1)/(sqrt(8t-3))
12) g(t)= ((4t-3)^2)/ ((5t+1)^4)
Thanks.
To find the derivatives of the given functions, we can use the power rule and the chain rule.
11) Given function: f(t) = 1 / √(8t - 3)
Let's use the chain rule here. The chain rule states that if we have a function g(h(t)), the derivative of g(h(t)) with respect to t is g'(h(t)) * h'(t).
First, let's find the derivative of the inside function h(t) = 8t - 3:
h'(t) = 8
Next, let's find the derivative of the outside function g(u) = 1 / √u:
g'(u) = -1 / (2√u^3)
Now, we can apply the chain rule:
f'(t) = g'(h(t)) * h'(t)
= -1 / (2√(8t - 3)^3) * 8
= -4 / (√(8t - 3)^3)
So, the derivative of f(t) = 1 / √(8t - 3) is f'(t) = -4 / (√(8t - 3)^3).
12) Given function: g(t) = ((4t - 3)^2) / ((5t + 1)^4)
Let's use the quotient rule here. The quotient rule states that if we have a function f(t) = g(t) / h(t), the derivative of f(t) with respect to t is [h(t) * g'(t) - g(t) * h'(t)] / (h(t))^2.
First, let's find the derivative of the numerator function g(t) = (4t - 3)^2:
g'(t) = 2(4t - 3) * 4
= 8(4t - 3)
Now, let's find the derivative of the denominator function h(t) = (5t + 1)^4:
h'(t) = 4(5t + 1)^3 * 5
= 20(5t + 1)^3
Now, we can apply the quotient rule:
g'(t) = [h(t) * g'(t) - g(t) * h'(t)] / (h(t))^2
= [(5t + 1)^4 * 8(4t - 3) - (4t - 3)^2 * 20(5t + 1)^3] / [(5t + 1)^4]^2
= [8(5t + 1)^4(4t - 3) - 20(4t - 3)^2(5t + 1)^3] / [(5t + 1)^4]^2
So, the derivative of g(t) = ((4t - 3)^2) / ((5t + 1)^4) is g'(t) = [8(5t + 1)^4(4t - 3) - 20(4t - 3)^2(5t + 1)^3] / [(5t + 1)^4]^2.