Question

if $f(x)=5x^{2}-x^{3}$, find $f(1)$ and use it to find an equation of the tangent line to the curve $y = 5x^{2}-x^{3}$ at the point $(1,4)$.

Explanation:

Step1: Find the derivative of $f(x)$

Using the power - rule $\frac{d}{dx}(x^n)=nx^{n - 1}$, if $f(x)=5x^{2}-x^{3}$, then $f'(x)=\frac{d}{dx}(5x^{2}-x^{3})=5\times2x-3x^{2}=10x - 3x^{2}$.

Step2: Evaluate $f'(1)$

Substitute $x = 1$ into $f'(x)$. So $f'(1)=10\times1-3\times1^{2}=10 - 3=7$.

Step3: Use the point - slope form to find the tangent line equation

The point - slope form of a line is $y - y_{1}=m(x - x_{1})$, where $(x_{1},y_{1})=(1,4)$ is the point on the curve and $m = f'(1)=7$.
Substitute these values into the formula: $y - 4=7(x - 1)$.
Expand the right - hand side: $y-4 = 7x-7$.
Add 4 to both sides to get the equation of the tangent line: $y=7x - 3$.

Answer:

$f'(1)=7$, and the equation of the tangent line is $y = 7x-3$