Can somebody explain to me the chain rule using this example (out of an old higher paper)? I'm looking up example online and im confused where they are getting some of the numbers from.
A) Given that y=(x^2 +7)^(1/2) find dy/dx
Find the derivative of the following via implicit differentiation:
d/dx(y) = d/dx(sqrt(7+x^2))
The derivative of y is y'(x):
y'(x) = d/dx(sqrt(7+x^2))
Using the chain rule, d/dx(sqrt(x^2+7)) = ( dsqrt(u))/( du) 0, where u = x^2+7 and ( d)/( du)(sqrt(u)) = 1/(2 sqrt(u)):
y'(x) = (d/dx(7+x^2))/(2 sqrt(7+x^2))
Differentiate the sum term by term:
y'(x) = d/dx(7)+d/dx(x^2)/(2 sqrt(7+x^2))
The derivative of 7 is zero:
y'(x) = (d/dx(x^2)+0)/(2 sqrt(7+x^2))
Simplify the expression:
y'(x) = (d/dx(x^2))/(2 sqrt(7+x^2))
Use the power rule, d/dx(x^n) = n x^(n-1), where n = 2: d/dx(x^2) = 2 x:
y'(x) = 2 x/(2 sqrt(7+x^2))
Simplify the expression:
y'(x) = x/sqrt(7+x^2)
Expand the left hand side:
Answer: |y'(x) = x/sqrt(7+x^2)
+129840 A) Given that y=(x^2 +7)^(1/2) find dy/dx
Let x^2 + 7 = m
And we can write
dy/dx = dy/dm * dm/dx
So we have
y = m^(1/2) taking the derivative of this =
dy/dm = (1/2)m^(-1/2)
And
dm/dx = 2x
So...puting this all together, we have
dy/dx = dy/dm * dm/dx = (1/2)m^(-1/2) * 2x = (1/2)(x^2 + 7)^(-1/2) * 2x = x * (x^2 + 7)^(-1/2)
