Find the Tangent Line at x=4 f(x)=x^2+5 , x=4

Substitute ${\displaystyle 4}$ in for ${\displaystyle x}$.

Solve for ${\displaystyle y}$.

By the Sum Rule, the derivative of ${\displaystyle x^2+5}$ with respect to ${\displaystyle x}$ is ${\displaystyle \frac{d}{dx}\left[x^2\right]+\frac{d}{dx}\left[5\right]}$.

Differentiate using the Power Rule which states that ${\displaystyle \frac{d}{dx}\left[x^n\right]}$ is ${\displaystyle n{x}^{n-1}}$ where ${\displaystyle n=2}$.

Since ${\displaystyle 5}$ is constant with respect to ${\displaystyle x}$, the derivative of ${\displaystyle 5}$ with respect to ${\displaystyle x}$ is ${\displaystyle 0}$.

Add ${\displaystyle 2x}$ and ${\displaystyle 0}$.

Evaluate the derivative at ${\displaystyle x=4}$.

Multiply ${\displaystyle 2}$ by ${\displaystyle 4}$.

Use the slope ${\displaystyle 8}$ and a given point ${\displaystyle (4,21)}$ to substitute for ${\displaystyle x_1}$ and ${\displaystyle y_1}$ in the point-slope form ${\displaystyle y-y_1=m(x-x_1)}$, which is derived from the slope equation ${\displaystyle m=\frac{y_2-y_1}{x_2-x_1}}$.

Simplify the equation and keep it in point-slope form.

Solve for ${\displaystyle y}$.