When you do implicit differentiation, how does D = √(x^2 + 8x + 12) turn into dD/dt = [(x + 4)(dx/dt)]/√(x^2 + 8x + 12)?
please explain...I don't even understand where the (x+4) comes from. D means distance, but that's irrelevant
It's really just the chain rule in disguise:
D = √u
where u is a function of x
dD/dx = 1/(2√u) du/dx
du/dx = 2x+8
Now, since x is a function of t,
dD/dt = dD/dx dx/dt
and voila
D = √(x^2 + 8x + 12)
We can also rewrite this as
D = (x^2 + 8x + 12)^(1/2)
Now we have to get dD/dt. To get the derivative, recall that if x is raised to a certain constant, the derivative is
x^n = n*x^(n-1) * dx
We multiply the exponent to x, which is jow raised to 1 less than the original exponent, and multiplied by the derivative of x.
For example,
derivative of (2x)^5 = 5*(3x)^4 * 3 = 15*(3x)^4
Don't forget that you have to multiply the derivative of 3x (which is 3) to the whole expression.
Therefore, in the problem, dD/dt((x^2 + 8x + 12)^(1/2))
= ((1/2)*(x^2 + 8x + 12)^(1/2 - 1) * d/dx((x^2 + 8x + 12)))*dx/dt
We know that the derivative of the x^2 + 8x + 12 is 2x + 8:
= (1/2)*(2x + 8)*(x^2 + 8x + 12)^(-1/2) dx/dt
= (x + 4)*(x^2 + 8x + 12)^(-1/2) dx/dt
Note that a term raised by negative exponent can be rewritten as a term raised with positive exponent, but now placed in the denominator. Rewriting this expression, you'll get the answer you typed there:
= [(x + 4)/√(x^2 + 8x + 12)](dx/dt)
Hope this helps :3
D^2 = (x^2 + 8x + 12)
2D dD/dt = 2x dx/dt + 8 dx/dt
dD/dt = dx/dt (2x+8)/2D
dD/dt = dx/dt (x+4) / √(x^2 + 8x + 12)
Nice one, Reiny. Went right to the heart of the matter of implicit derivatives.
Thank you all so much!
To understand how the expression D = √(x^2 + 8x + 12) turns into dD/dt = [(x + 4)(dx/dt)]/√(x^2 + 8x + 12), we need to apply the chain rule of differentiation.
The chain rule states that for a composite function y = f(g(x)), the derivative dy/dx can be found by multiplying the derivative of the outer function f'(g(x)) with the derivative of the inner function g'(x). In this case, D is a function of x, and we want to find its derivative with respect to t (dt). So we can express D as D(x), and we want to find dD/dt.
Let's break down the steps for this problem:
1. Start with the equation D = √(x^2 + 8x + 12).
2. Take the derivative of both sides with respect to t (dt):
dD/dt = d/dt(√(x^2 + 8x + 12))
3. Apply the chain rule by recognizing that D is a function of x:
dD/dt = dD/dx * dx/dt
4. Find dD/dx, the derivative of D with respect to x:
dD/dx = d/dx(√(x^2 + 8x + 12))
5. To differentiate √(x^2 + 8x + 12) with respect to x, we rewrite it as (x^2 + 8x + 12)^(1/2):
dD/dx = d/dx((x^2 + 8x + 12)^(1/2))
6. Using the power rule, we differentiate (x^2 + 8x + 12)^(1/2) with respect to x:
dD/dx = (1/2)(x^2 + 8x + 12)^(-1/2) * d/dx(x^2 + 8x + 12)
7. Simplify d/dx(x^2 + 8x + 12) to find the derivative of the expression:
dD/dx = (1/2)(x^2 + 8x + 12)^(-1/2) * (2x + 8)
8. Next, rewrite (x^2 + 8x + 12)^(-1/2) as 1/√(x^2 + 8x + 12):
dD/dx = (1/2)(2x + 8)/√(x^2 + 8x + 12)
9. Now, substitute this result back into the chain rule expression:
dD/dt = (1/2)(2x + 8)/√(x^2 + 8x + 12) * dx/dt
10. Simplify the expression by canceling out the 2:
dD/dt = (x + 4)/√(x^2 + 8x + 12) * dx/dt
Therefore, the final expression is dD/dt = [(x + 4)(dx/dt)]/√(x^2 + 8x + 12), with the factor (x + 4) resulting from the application of the chain rule.