Questions: For the function f(x)=7x^2-3x+5x^3,
Step 2 of 2: Find the equation of the tangent line at x=4.
Transcript text: For the function $f(x)=7 x^{2}-3 x+5 x^{3}$,
Step 2 of 2: Find the equation of the tangent line at $x=4$.
Solution
Solution Steps
Step 1: Find the Derivative
The function is given by \( f(x) = 5x^3 + 7x^2 - 3x \). To find the slope of the tangent line, we first calculate the derivative:
\[
f'(x) = 15x^2 + 14x - 3
\]
Step 2: Evaluate the Derivative at \( x = 4 \)
Next, we evaluate the derivative at \( x = 4 \) to find the slope of the tangent line:
\[
f'(4) = 15(4^2) + 14(4) - 3 = 293
\]
Step 3: Calculate the Function Value at \( x = 4 \)
We then calculate the value of the function at \( x = 4 \):
\[
f(4) = 5(4^3) + 7(4^2) - 3(4) = 420
\]
Step 4: Write the Equation of the Tangent Line
Using the point-slope form of the equation of a line, \( y - y_1 = m(x - x_1) \), where \( m = 293 \), \( x_1 = 4 \), and \( y_1 = 420 \), we can express the equation of the tangent line:
\[
y - 420 = 293(x - 4)
\]
Simplifying this gives:
\[
y = 293x - 752
\]
Final Answer
The equation of the tangent line at \( x = 4 \) is
\[
\boxed{y = 293x - 752}
\]