find the slope of the tangent line to the graph of the given function at the given value of x. find the…

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 y = x^4 - 5x^3 + 9; x = 2\nhow would the slope of a tangent line be determined with the given information?\na. substitute values of x into the equation and solve for y. plot the resulting points to find the linear equation.\nb. substitute values of y into the equation and solve for x. plot the resulting points to find the linear equation.\nc. set the derivative equal to zero and solve for x.\nd. substitute 2 for x into the derivative of the function and evaluate.\nthe slope of the tangent line is - 28.\nthe equation of the line is . (type an equation. type your answer in slope - intercept form.)

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 y = x^4 - 5x^3 + 9; x = 2\nhow would the slope of a tangent line be determined with the given information?\na. substitute values of x into the equation and solve for y. plot the resulting points to find the linear equation.\nb. substitute values of y into the equation and solve for x. plot the resulting points to find the linear equation.\nc. set the derivative equal to zero and solve for x.\nd. substitute 2 for x into the derivative of the function and evaluate.\nthe slope of the tangent line is - 28.\nthe equation of the line is . (type an equation. type your answer in slope - intercept form.)

Answer

Explanation:

Step1: Find the derivative of the function

The power - rule for differentiation is $\frac{d}{dx}(x^n)=nx^{n - 1}$. Given $y=x^{4}-5x^{3}+9$, then $y^\prime=\frac{d}{dx}(x^{4}-5x^{3}+9)=4x^{3}-15x^{2}$.

Step2: Evaluate the derivative at $x = 2$

Substitute $x = 2$ into $y^\prime$. $y^\prime(2)=4\times2^{3}-15\times2^{2}=4\times8 - 15\times4=32-60=-28$. This is the slope of the tangent line.

Step3: Find the $y$-coordinate when $x = 2$

Substitute $x = 2$ into the original function $y=x^{4}-5x^{3}+9$. $y(2)=2^{4}-5\times2^{3}+9=16 - 40 + 9=-15$.

Step4: Use the point - slope form to find the equation of the tangent line

The point - slope form of a line is $y - y_1=m(x - x_1)$, where $(x_1,y_1)=(2,-15)$ and $m=-28$. So $y+15=-28(x - 2)$.

Step5: Rewrite in slope - intercept form

Expand the right - hand side: $y+15=-28x + 56$. Then $y=-28x+41$.

Answer:

$y=-28x + 41$