if ( y = (x^{2}+9)^{4} ), find ( \frac{d^{2}y}{dx^{2}} )\n( \frac{d^{2}y}{dx^{2}}=square )
if ( y = (x^{2}+9)^{4} ), find ( \frac{d^{2}y}{dx^{2}} )\n( \frac{d^{2}y}{dx^{2}}=square )
Answer
Explanation:
Step1: Find the first - derivative using the chain rule
The chain rule states that if (y = u^n) where (u) is a function of (x), then (\frac{dy}{dx}=n\cdot u^{n - 1}\cdot\frac{du}{dx}). Let (u=x^{2}+9) and (n = 4). Then (\frac{du}{dx}=2x). So, (\frac{dy}{dx}=4(x^{2}+9)^{3}\cdot(2x)=8x(x^{2}+9)^{3}).
Step2: Find the second - derivative using the product rule
The product rule states that if (y = f(x)\cdot g(x)), then (\frac{dy}{dx}=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)). Let (f(x)=8x) and (g(x)=(x^{2}+9)^{3}). First, find (f^{\prime}(x)) and (g^{\prime}(x)): (f^{\prime}(x)=8). For (g^{\prime}(x)), use the chain rule again. Let (v=x^{2}+9), (m = 3). Then (\frac{dv}{dx}=2x), and (g^{\prime}(x)=3(x^{2}+9)^{2}\cdot(2x)=6x(x^{2}+9)^{2}). Now, by the product rule: (\frac{d^{2}y}{dx^{2}}=8\cdot(x^{2}+9)^{3}+8x\cdot6x(x^{2}+9)^{2}). Factor out (8(x^{2}+9)^{2}): (\frac{d^{2}y}{dx^{2}}=8(x^{2}+9)^{2}[(x^{2}+9)+6x^{2}]). Simplify the expression inside the brackets: ((x^{2}+9)+6x^{2}=7x^{2}+9).
Answer:
(8(x^{2}+9)^{2}(7x^{2}+9))