Question

given y = f(u) and u = g(x), find $\frac{dy}{dx}=f(g(x))g(x)$. y = 7u^7, u = 4x + 17 $\frac{dy}{dx}=square$

Explanation:

Step1: Find the derivative of y with respect to u

Using the power - rule for differentiation, if $y = 7u^{7}$, then $y^\prime=f^\prime(u)=\frac{d}{du}(7u^{7})$. The power - rule states that $\frac{d}{du}(au^{n})=nau^{n - 1}$, so $f^\prime(u)=7\times7u^{6}=49u^{6}$.

Step2: Find the derivative of u with respect to x

If $u = 4x+17$, then $u^\prime = g^\prime(x)=\frac{d}{dx}(4x + 17)$. Since $\frac{d}{dx}(ax + b)=a$ for constants $a$ and $b$, $g^\prime(x)=4$.

Step3: Apply the chain - rule

The chain - rule states that $\frac{dy}{dx}=f^\prime(g(x))g^\prime(x)$. Substitute $u = g(x)=4x + 17$ into $f^\prime(u)$ and multiply by $g^\prime(x)$. So $\frac{dy}{dx}=49(4x + 17)^{6}\times4$.

Step4: Simplify the result

$49\times4(4x + 17)^{6}=196(4x + 17)^{6}$.

Answer:

$196(4x + 17)^{6}$