Use the power rule to find the derivative.

Remember the power rule is:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

\(\displaystyle \frac{d}{dx}2x^6=12x^5\)

\(\displaystyle \frac{d}{dx}-3x^3=-9x^2\)

\(\displaystyle \frac{d}{dx}2x=2\)

Thus, the derivative is \(\displaystyle 12x^5-9x^2+2\)

First, find the derivative using the power rule

Recall the power rule:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

\(\displaystyle \frac{d}{dx}4x=4\)

\(\displaystyle \frac{d}{dx}-5=0\)

Because the derivative is a constant \(\displaystyle 4\), then the derivative at all points is \(\displaystyle 4\).

First, find the derivative using the power rule. 

Recall the power rule:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

\(\displaystyle \frac{d}{dx}x^2=2x\)

\(\displaystyle \frac{d}{dx}9x=9\)

Recall that the derivative of a constant is zero.

The derivative is \(\displaystyle 2x+9.\)

Now, substitute \(\displaystyle 0\) for \(\displaystyle x\).

\(\displaystyle 2(0)+9\)

\(\displaystyle =9\)

Uses the product rule to find this derivative.

Recall the product rule:

\(\displaystyle \\ \\ f(x)=(f(x)g(x))' \\ f'(x)=f'(x)g(x)+f(x)g'(x)\)

\(\displaystyle f'(x)=2x\sin (x)+x^2\cos (x)\)

Use the product rule to find this derivative.

Recall the product rule:

\(\displaystyle \\ \\ f(x)=(f(x)g(x))' \\ f'(x)=f'(x)g(x)+f(x)g'(x)\)

\(\displaystyle f'(x)=5\cos (x)-5x\sin (x)\)

First, find the derivative using the power rule. 

Recall the power rule:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

\(\displaystyle \frac{d}{dx}5x^2=10x\)

\(\displaystyle \frac{d}{dx}-7=0\)

The derivative is \(\displaystyle 10x\).

Now, substitute \(\displaystyle 8\) for \(\displaystyle x\).  The slope of the tangent line at \(\displaystyle x=8\) is \(\displaystyle 80\).

Use the quotient rule to find this derivative.

Recall the quotient rule:

\(\displaystyle \\ \\ h(x)=\frac{f(x)}{g(x)} \\ \\ h'(x)=\frac{g(x)f'(x)-g'(x)f(x)}{(g(x))^2}\)

\(\displaystyle h'(x)=\frac{x^4(0)-2(4x^3)}{-x^8}\)

\(\displaystyle =\frac{-8x^3}{-x^8}=\frac{8x^3}{x^8}=\frac{8}{x^5}\)

Begin by simplifying the expression.

\(\displaystyle x^2+5x-4\)

Now, find the derivative using the power rule.

Recall the power rule:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

\(\displaystyle 2x+5\)

Finally, substitute \(\displaystyle 7\) for \(\displaystyle x\).

\(\displaystyle 2(7)+5=19\)

First, find the derivative using the power rule. 

Recall the power rule:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

\(\displaystyle \frac{d}{dx}x^3=3x^2\)

Now, substitute \(\displaystyle 6\) for \(\displaystyle x\).

\(\displaystyle 3(6^2)=3(36)=108\)

Use the power rule to find the derivative.

Recall the power rule:

\(\displaystyle \\ \\ f(x)=ax^n \\ f'(x)=nax^{n-1}\)

\(\displaystyle \frac{d}{dx}5x^7=35x^6\)