suppose that a dimension x and the area a = 4x² of a shape are differentiable functions of t. write an…

suppose that a dimension x and the area a = 4x² of a shape are differentiable functions of t. write an equation that relates (\frac{da}{dt}) to (\frac{dx}{dt}). (\frac{da}{dt}=square)

suppose that a dimension x and the area a = 4x² of a shape are differentiable functions of t. write an equation that relates (\frac{da}{dt}) to (\frac{dx}{dt}). (\frac{da}{dt}=square)

Answer

Explanation:

Step1: Differentiate (A = 4x^{2}) with respect to (t)

Use the chain - rule (\frac{dA}{dt}=\frac{dA}{dx}\cdot\frac{dx}{dt}). First, find (\frac{dA}{dx}) for (A = 4x^{2}). By the power rule (\frac{d}{dx}(ax^{n})=nax^{n - 1}), so (\frac{dA}{dx}=\frac{d}{dx}(4x^{2})=8x).

Step2: Apply the chain - rule formula

Since (\frac{dA}{dt}=\frac{dA}{dx}\cdot\frac{dx}{dt}) and (\frac{dA}{dx}=8x), then (\frac{dA}{dt}=8x\frac{dx}{dt}).

Answer:

(8x\frac{dx}{dt})