1.Find the derivative of the function.
f(t) = (1+squrt t)(8t^2-7)
2. Find the derivative of the function.
f(x) = (x^3 - 4)/(x^2 + 4)
To find the derivative of a function, you can use the rules of differentiation. Here's how you can find the derivatives of the given functions:
1. Derivative of f(t) = (1 + sqrt(t))(8t^2 - 7):
To find the derivative, we can use the product rule, which states that for two functions u(t) and v(t), the derivative of their product is given by the formula:
(d/dt)(u(t) * v(t)) = u'(t) * v(t) + u(t) * v'(t),
where u'(t) and v'(t) are the derivatives of u(t) and v(t) with respect to t.
Let's apply the product rule to find the derivative of f(t):
u(t) = (1 + sqrt(t))
v(t) = (8t^2 - 7)
First, find the derivatives of u(t) and v(t):
u'(t) = d/dt(1 + sqrt(t)) = 0 + (1/2)(1/sqrt(t)) = 1/2sqrt(t)
v'(t) = d/dt(8t^2 - 7) = 16t
Now, apply the product rule:
f'(t) = u'(t) * v(t) + u(t) * v'(t)
= (1/2sqrt(t)) * (8t^2 - 7) + (1 + sqrt(t)) * (16t)
= (4t - 7/2sqrt(t)) + (16t + 16t*sqrt(t))
= 20t + 16t*sqrt(t) - 7/2sqrt(t)
Therefore, the derivative of f(t) is f'(t) = 20t + 16t*sqrt(t) - 7/2sqrt(t).
2. Derivative of f(x) = (x^3 - 4)/(x^2 + 4):
To find the derivative of this function, we can use the quotient rule, which states that for two functions u(x) and v(x), the derivative of their quotient is given by the formula:
(d/dx)(u(x) / v(x)) = (u'(x) * v(x) - u(x) * v'(x)) / (v(x))^2,
where u'(x) and v'(x) are the derivatives of u(x) and v(x) with respect to x.
Let's apply the quotient rule to find the derivative of f(x):
u(x) = (x^3 - 4)
v(x) = (x^2 + 4)
First, find the derivatives of u(x) and v(x):
u'(x) = d/dx(x^3 - 4) = 3x^2
v'(x) = d/dx(x^2 + 4) = 2x
Now, apply the quotient rule:
f'(x) = (u'(x) * v(x) - u(x) * v'(x)) / (v(x))^2
= (3x^2 * (x^2 + 4) - (x^3 - 4) * 2x) / (x^2 + 4)^2
= (3x^4 + 12x^2 - 2x^4 + 8x) / (x^2 + 4)^2
= (x^4 + 12x^2 + 8x) / (x^2 + 4)^2
Therefore, the derivative of f(x) is f'(x) = (x^4 + 12x^2 + 8x) / (x^2 + 4)^2.