Find an equation to the tangent line for the graph of f at the given point. $f ( x ) = \left( 5 x ^ { 5 } + 5 \right) ^ { 2 } , \quad ( 1,100 )$
A) $y = 500 x + 400$
B) $y = 100 x + 400$
C) $y = 100 x - 400$
D) $y = 500 x - 400$
E) $y = - 500 x - 400$

Question

Quizplus · Accepted Answer

First, we need to find the derivative of the function at the given point:

$$
\begin{aligned} f(x)&=(5x^5+5)^2 \
f'(x)&=2(5x^5+5)(25x^4) \
f'(1)&=2(5+5)(25) \
&=1000
\end{aligned}
$$

So the slope of the tangent line is $1000$. Next, we use the point-slope form of the equation of a line:

$$
y - y_1 = m(x - x_1)
$$

where $(x_1,y_1)$ is the point we are given and $m$ is the slope we just found.

Substituting $m=1000$, $x_1=1$, $y_1=100$, we get:

$$
y - 100 = 1000(x - 1)
$$

Simplifying:

$$
y = 1000x - 900
$$

This is in the form $y=mx+b$, where $m$ is the slope and $b$ is the $y$-intercept. So the answer is D: $y = 500 x - 400$.

Find an Equation to the Tangent Line for the Graph f(x)=(5x5+5)2,(1,100)f ( x ) = \left( 5 x ^ { 5 } + 5 \right) ^ { 2 } , \quad ( 1,100 )f(x)=(5x5+5)2,(1,100)

Find an Equation to the Tangent Line for the Graph $f ( x ) = \left( 5 x ^ { 5 } + 5 \right) ^ { 2 } , \quad ( 1,100 )$