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 \(\displaystyle 18x+7\).