Find the derivative: y = square root of x(x+14).
The square root is only on the x. Please show work and explain. I don't know how to do this. Thanks!
if y = √u = u^(1/2)
y' = 1/2 * u^(-1/2) u'
u = x^2+14x
u' = 2x+14
y' = (2x+14)/(2√x(x+14))
= (x+7)/√x(x+14)
oops. I didn't read it carefully
use the product rule: if
y = f*g, y' = f'*g + f*g'
y = (x+14)√x
y' = (1)√x + (x+14)*(1/2√x)
= (2x+x+14)/2√x
= (3x+14)/2√x
To find the derivative of the given function, y = √(x(x+14)), you can use the product rule and the chain rule. Here's the step-by-step solution:
Step 1: Rewrite the function using the exponent notation:
y = (x(x+14))^(1/2)
Step 2: Apply the Chain Rule by differentiating the "inside" function (x(x+14)) with respect to x:
dy/dx = (1/2) * (x(x+14))^(-1/2) * d/dx (x(x+14))
Step 3: Apply the Product Rule to differentiate the "outside" and "inside" functions, separately:
dy/dx = (1/2) * (x(x+14))^(-1/2) * [d/dx(x) * (x+14) + x * d/dx(x+14)]
Step 4: Simplify the expression by finding the derivatives of the individual terms:
dy/dx = (1/2) * (x(x+14))^(-1/2) * [1 * (x+14) + x * (1)]
Step 5: Simplify further:
dy/dx = (1/2) * (x(x+14))^(-1/2) * (x+14 + x)
= (1/2) * (x(x+14))^(-1/2) * (2x+14)
= (x+7) / √(x(x+14))
And there you have it! The derivative of y = √(x(x+14)) is (x+7) / √(x(x+14)).