Find the derivative of the function.
g(t) = ((8t + 10)^2)*(((10t^2) - 7)^-3)
I am supposed to use The Chain Rule. However, I am confused as to how to do this.
To find the derivative of the given function using the Chain Rule, follow these steps:
Step 1: Identify the inner function and the outer function.
The inner function in this case is (10t^2) - 7, and the outer function is ((8t + 10)^2).
Step 2: Determine the derivatives of the inner and outer functions.
The derivative of the inner function (10t^2) - 7 is:
d/dt [ (10t^2) - 7 ] = 20t
The derivative of the outer function ((8t + 10)^2) can be found using the power rule and the chain rule.
Let u = 8t + 10.
Using the chain rule, the derivative of the outer function is:
d/du [ (u)^2 ] * d/dt [ 8t + 10 ]
= 2u * 8
= 16u
Step 3: Apply the Chain Rule.
Now, apply the Chain Rule by multiplying the derivatives of the inner and outer functions.
d/dt [ ((8t + 10)^2) * ((10t^2) - 7)^-3 ]
= 16u * 20t * ((10t^2) - 7)^-3
Step 4: Convert back to t.
Since we expressed the derivative in terms of u, substitute the value of u back into the expression:
= 16(8t + 10) * 20t * ((10t^2) - 7)^-3
Simplifying this expression further is possible but requires additional computations based on the given function.