Question

find dy.
y = 26 + 7x - 5x^3
dy = (simplify your answer.)

Explanation:

Step1: Recall derivative rules

The derivative of a constant $C$ is $0$, i.e., $\frac{dC}{dx}=0$; the derivative of $ax^n$ is $nax^{n - 1}$ by the power - rule $\frac{d(ax^n)}{dx}=nax^{n - 1}$.

Step2: Differentiate each term

For $y = 26+7x - 5x^{3}$, the derivative of $26$ with respect to $x$ is $\frac{d(26)}{dx}=0$, the derivative of $7x$ with respect to $x$ is $\frac{d(7x)}{dx}=7$ (since $n = 1$ and $a = 7$, so $1\times7x^{1 - 1}=7$), and the derivative of $-5x^{3}$ with respect to $x$ is $\frac{d(-5x^{3})}{dx}=-15x^{2}$ (since $n = 3$ and $a=-5$, so $3\times(-5)x^{3 - 1}=-15x^{2}$).

Step3: Find $dy$

We know that $dy=y'dx$. Since $y'=\frac{dy}{dx}=0 + 7-15x^{2}=7 - 15x^{2}$, then $dy=(7 - 15x^{2})dx$.

Answer:

$(7 - 15x^{2})dx$