for this assignment to count as extra credit, it must be submitted online by the due date/time at the end of…

for this assignment to count as extra credit, it must be submitted online by the due date/time at the end of the week with the si leaders signature at the bottom. also, before leaving the si session, be sure to ask your si leader for clarification on any topics covered thus far in your class and any questions you have pertaining to homework problems. 1. complete each row with what should follow the equal sign. derivative rules constant, c $\frac{d}{dx}(c)=$ power $\frac{d}{dx}(x^{n})=$ power with constant, c $\frac{d}{dx}(cx^{n})=$ sum/difference $\frac{d}{dx}f(x)pm g(x)=$ product $\frac{d}{dx}(f(x))(g(x))=$ quotient $\frac{d}{dx}(\frac{f(x)}{g(x)})=$ natural exponential function $\frac{d}{dx}(e^{x})=$ 2. now, recall these helpful laws of exponents. replace the question marks to make true statements. laws of exponents? provided $a\neq0$, $1 = a^{?}$ provided $a\neq0$, $\frac{1}{a^{n}}=a^{?}$ provided $a\neq0$, $\frac{a^{m}}{a^{n}}=a^{?}$ provided $a\neq0$, express $(\frac{bc}{a})^{n}$ without parentheses $(\frac{bc}{a})^{n}=?$ $a = a^{?}$ $a^{m}a^{n}=a^{?}$ $(a^{m})^{n}=a^{?}$ $sqrtn{a^{m}}=a^{?}$

for this assignment to count as extra credit, it must be submitted online by the due date/time at the end of the week with the si leaders signature at the bottom. also, before leaving the si session, be sure to ask your si leader for clarification on any topics covered thus far in your class and any questions you have pertaining to homework problems. 1. complete each row with what should follow the equal sign. derivative rules constant, c $\frac{d}{dx}(c)=$ power $\frac{d}{dx}(x^{n})=$ power with constant, c $\frac{d}{dx}(cx^{n})=$ sum/difference $\frac{d}{dx}f(x)pm g(x)=$ product $\frac{d}{dx}(f(x))(g(x))=$ quotient $\frac{d}{dx}(\frac{f(x)}{g(x)})=$ natural exponential function $\frac{d}{dx}(e^{x})=$ 2. now, recall these helpful laws of exponents. replace the question marks to make true statements. laws of exponents? provided $a\neq0$, $1 = a^{?}$ provided $a\neq0$, $\frac{1}{a^{n}}=a^{?}$ provided $a\neq0$, $\frac{a^{m}}{a^{n}}=a^{?}$ provided $a\neq0$, express $(\frac{bc}{a})^{n}$ without parentheses $(\frac{bc}{a})^{n}=?$ $a = a^{?}$ $a^{m}a^{n}=a^{?}$ $(a^{m})^{n}=a^{?}$ $sqrtn{a^{m}}=a^{?}$

Answer

Explanation:

Step1: Constant rule

The derivative of a constant $c$ with respect to $x$ is 0. So $\frac{d}{dx}(c)=0$.

Step2: Power - rule

The derivative of $x^n$ with respect to $x$ is $nx^{n - 1}$. So $\frac{d}{dx}(x^n)=nx^{n - 1}$.

Step3: Power - rule with constant

Using the constant - multiple rule of differentiation, $\frac{d}{dx}(cx^n)=c\frac{d}{dx}(x^n)=cnx^{n - 1}$.

Step4: Sum/Difference rule

The derivative of the sum or difference of two functions is the sum or difference of their derivatives. So $\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime}(x)\pm g^{\prime}(x)$.

Step5: Product rule

The derivative of the product of two functions is given by $\frac{d}{dx}[(f(x))(g(x))]=f(x)g^{\prime}(x)+f^{\prime}(x)g(x)$.

Step6: Quotient rule

The derivative of the quotient of two functions is $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{g^{2}(x)}$, where $g(x)\neq0$.

Step7: Natural exponential function rule

The derivative of $e^x$ with respect to $x$ is $e^x$. So $\frac{d}{dx}(e^x)=e^x$.

For the laws of exponents:

Step8: $1 = a^0$ (Zero - exponent rule)

For any non - zero real number $a$, $a^0 = 1$.

Step9: $\frac{1}{a^n}=a^{-n}$ (Negative - exponent rule)

Step10: $\frac{a^m}{a^n}=a^{m - n}$ (Quotient rule of exponents)

Step11: $\left(\frac{bc}{a}\right)^n=\frac{b^n c^n}{a^n}$ (Power of a quotient rule)

Step12: $a=a^1$ (First - exponent rule)

Step13: $a^m a^n=a^{m + n}$ (Product rule of exponents)

Step14: $(a^m)^n=a^{mn}$ (Power of a power rule)

Step15: $\sqrt[n]{a^m}=a^{\frac{m}{n}}$ (Fractional - exponent rule)

Answer:

  1. $\frac{d}{dx}(c)=0$
    • $\frac{d}{dx}(x^n)=nx^{n - 1}$
    • $\frac{d}{dx}(cx^n)=cnx^{n - 1}$
    • $\frac{d}{dx}[f(x)\pm g(x)]=f^{\prime}(x)\pm g^{\prime}(x)$
    • $\frac{d}{dx}[(f(x))(g(x))]=f(x)g^{\prime}(x)+f^{\prime}(x)g(x)$
    • $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{f^{\prime}(x)g(x)-f(x)g^{\prime}(x)}{g^{2}(x)}$
    • $\frac{d}{dx}(e^x)=e^x$
    • $1 = a^0$
    • $\frac{1}{a^n}=a^{-n}$
    • $\frac{a^m}{a^n}=a^{m - n}$
    • $\left(\frac{bc}{a}\right)^n=\frac{b^n c^n}{a^n}$
    • $a=a^1$
    • $a^m a^n=a^{m + n}$
    • $(a^m)^n=a^{mn}$
    • $\sqrt[n]{a^m}=a^{\frac{m}{n}}$