Use the power rule to find the derivative.

The power rule states,

$\displaystyle \frac{d}{dx}x^n=nx^{n-1}$.

Applying this rule to the function in the problem results in the following.

$\displaystyle \frac{d}{dx}(-4x^3)=3(-4)x^{3-1}=-12x^2$

Use the power rule to find the derivative. 

The power rule states,

$\displaystyle \frac{d}{dx}x^n=nx^{n-1}$.

Applying this rule to each term of the function results in the following.

$\displaystyle \frac{d}{dx}4x=4$

$\displaystyle \frac{d}{dx}-3=0$

Thus, the derivative is 4.

To find the equation for the slope of the tangent line, find the derivative.

To find the derivative, use the power rule.

The power rule states,

$\displaystyle \frac{d}{dx}x^n=nx^{n-1}$.

Applying the power rule to each term in the function results in,

$\displaystyle \frac{d}{dx}8x^2=16x$

$\displaystyle \frac{d}{dx}-4x=-4$

$\displaystyle \frac{d}{dx}4=0$.

Thus, the derivative is $\displaystyle 16x-4$.

Use the power rule to find the derivative.

The power rule states,

$\displaystyle \frac{d}{dx}x^n=nx^{n-1}$.

Applying the power rule to each term within the function results in the following.

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

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

Thus, the derivative is $\displaystyle 3x^2+10x$

Now, substitute $\displaystyle 3$ for $\displaystyle x$.

$\displaystyle 3(3^2)+10(3)=57$

First, use the power rule to find the derivative.

The power rule states,

$\displaystyle \frac{d}{dx}x^n=nx^{n-1}$.

Applying the power rule to each term in the function results in the following.

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

$\displaystyle \frac{d}{dx}-3x=-3$

Thus, the derivative is $\displaystyle 10x-3$.

Now, substitute 2 for x.

$\displaystyle 10(2)-3=17$.

Use the product rule to find the derivative.

The product rule states,

$\displaystyle \frac{d}{dx}f(x)g(x)=f(x)g'(x)+g(x)f'(x)$.

Given,

$\displaystyle f(x)=2x$

$\displaystyle g(x)=cos(x)$

and recalling the trigonometry derivative for cosine is,

$\displaystyle \frac{d}{dx}cos(x)=-sin(x)$

the derivatives are as follows.

$\displaystyle f'(x)=1\cdot 2x^{1-1}=2$

$\displaystyle g'(x)=-sin(x)$

Therefore, using the product rule the derivative becomes,

$\displaystyle 2x(-\sin x)+2\cos (x)$.

Use the power rule to find the derivative.

The power rule states,

$\displaystyle \frac{d}{dx}x^n=nx^{n-1}$.

Applying the power rule to each term in the function results in the following.

$\displaystyle \frac{d}{dx}2x^4=8x^3$

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

Thus, the derivative is $\displaystyle 8x^3-3x^2$.

Use the product rule to find the derivative.

The product rule states,

$\displaystyle \frac{d}{dx}f(x)g(x)=f(x)g'(x)+g(x)f'(x)$.

Given,

$\displaystyle f(x)=5x$

$\displaystyle g(x)=sin(x)$

and recalling the trigonometry derivative for sine is,

$\displaystyle \frac{d}{dx}sin(x)=cos(x)$

the derivative becomes,

$\displaystyle 5x\cos (x)+5\sin (x)$.

First, find the derivative using the power rule.

The power rule states,

$\displaystyle \frac{d}{dx}x^n=nx^{n-1}$.

Applying the power rule results in the following.

$\displaystyle \frac{d}{dx}4x^2=8x$

Now, substitute 6 for x.

$\displaystyle 8(6)=48$.

Use the power rule to find the derivative. 

The power rule states,

$\displaystyle \frac{d}{dx}x^n=nx^{n-1}$.

Applying the power rule to each term in the function results in the following.

$\displaystyle \frac{d}{dx}9x^2=18x$

$\displaystyle \frac{d}{dx}7x=7$

$\displaystyle \frac{d}{dx}(-8)=0$

Thus, the derivative is $\displaystyle 18x+7$.