Question

find the slope of the tangent line to the graph of the given function at the given value of x. find the equation of the tangent line. y = x^4 - 5x^3 + 9; x = 2
how would the slope of a tangent line be determined with the given information?
a. substitute values of x into the equation and solve for y. plot the resulting points to find the linear equation.
b. substitute values of y into the equation and solve for x. plot the resulting points to find the linear equation.
c. set the derivative equal to zero and solve for x.
d. substitute 2 for x into the derivative of the function and evaluate.
the slope of the tangent line is □.

Explanation:

Step1: Find the derivative of the function

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, if $y=x^{4}-5x^{3}+9$, then $y^\prime = 4x^{3}-15x^{2}$.

Step2: Evaluate the derivative at $x = 2$

Substitute $x = 2$ into $y^\prime$:
$y^\prime(2)=4\times(2)^{3}-15\times(2)^{2}$
$=4\times8 - 15\times4$
$=32-60$
$=- 28$

Answer:

$-28$