Final answer to the problem
Step-by-step Solution
Specify the solving method
Multiply the single term $x-4$ by each term of the polynomial $\left(x+1\right)$
Multiply the single term $x$ by each term of the polynomial $\left(x-4\right)$
When multiplying two powers that have the same base ($x$), you can add the exponents
Combining like terms $-4x$ and $x$
Find the derivative of $x^2-3x-4$ using the definition. Apply the definition of the derivative: $\displaystyle f'(x)=\lim_{h\to0}\frac{f(x+h)-f(x)}{h}$. The function $f(x)$ is the function we want to differentiate, which is $x^2-3x-4$. Substituting $f(x+h)$ and $f(x)$ on the limit, we get
Expand $\left(x+h\right)^2$
Multiply the single term $-3$ by each term of the polynomial $\left(x+h\right)$
Solve the product $-\left(x^2-3x-4\right)$
Cancel like terms $x^2$ and $-x^2$
Multiply the single term $-1$ by each term of the polynomial $\left(-3x-4\right)$
Add the values $-4$ and $4$
Simplifying
Factor the polynomial $2xh+h^2-3h$ by it's greatest common factor (GCF): $h$
Simplify the fraction $\frac{h\left(2x+h-3\right)}{h}$ by $h$
Evaluate the limit $\lim_{h\to0}\left(2x+h-3\right)$ by replacing all occurrences of $h$ by $0$
Subtract the values $0$ and $-3$