Find the derivative of y = (x2 + 7x) / (x+2) using the quotient rule.

Question

ScanSolve AI Tutor · Accepted Answer

To find dy/dx for the given function, we will use the quotient rule. The function is y = (x² + 7x)/(x+2). Let u = x² + 7x and v = x+2. Step 1: Find the derivatives of u and v with respect to x. du/dx = d/dx(x² + 7x) = 2x + 7; dv/dx = d/dx(x+2) = 1; Step 2: Apply the quotient rule, which states that if y = u/v, then dy/dx = (v du/dx − u dv/dx)/(v²). dy/dx = ((x+2)(2x+7) − (x²+7x)(1))/((x+2)²) Step 3: Expand the numerator. (x+2)(2x+7) − (x²+7x)(1) = (2x² + 7x + 4x + 14) − (x² + 7x); = 2x² + 11x + 14 − x² − 7x Step 4: Combine like terms in the numerator. (2x² − x²) + (11x − 7x) + 14 = x² + 4x + 14 Step 5: Write the final expression for dy/dx. dy/dx = (x² + 4x + 14)/((x+2)²) The derivative is: dy/dx = (x² + 4x + 14)/((x+2)²)

Point	x	y
x-intercept	−7	0
x-intercept	0	0

Find the derivative of y = (x2 + 7x) / (x+2) using the quotient rule.

Answer

Step-by-step solution

Find the derivatives of $u$ and $v$ with respect to $x$ .

Apply the quotient rule, which states that if $y = \frac{u}{v}$ , then $\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$ .

Expand the numerator.

Combine like terms in the numerator.

Write the final expression for $\frac{dy}{dx}$ .

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Find the derivative of y = (x2 + 7x) / (x+2) using the quotient rule.

Answer

Step-by-step solution

Find the derivatives of uuu and vvv with respect to xxx.

Apply the quotient rule, which states that if y=uvy = \frac{u}{v}y=vu​, then dydx=vdudx−udvdxv2\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}dxdy​=v2vdxdu​−udxdv​​.

Expand the numerator.

Combine like terms in the numerator.

Write the final expression for dydx\frac{dy}{dx}dxdy​.

Try one yourself

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More Mathematics Questions

Find the derivatives of $u$ and $v$ with respect to $x$ .

Apply the quotient rule, which states that if $y = \frac{u}{v}$ , then $\frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^2}$ .

Write the final expression for $\frac{dy}{dx}$ .