\[ h(x)=x^{3}+2 x^{2}-x-8 \] (i) Find \( h^{\prime}(x) \), the derivative of \( h(x) \). \[ \begin{array}{l}h^{\prime}(x)=3 x^{2}+4 x^{1}-1 \\ \text { (ii) } \\ \text { Find the equation of the tangent to } h(x) \text { at the point }(2,6) \text {. } \\ h^{\prime}(x)=3 x^{2}+4 x-1 \\ h^{\prime}(2)=3(2)^{2}+1(2)-1 \\ 3(2)^{2}=3 \times 4=12 \\ 4(2)=8 \\ h^{\prime}(2)=12+81=19\end{array} \]

To find the equation of the tangent line at the point (2, 6) on the curve described by \( h(x) \), you can use the point-slope form of the equation of a line, which is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is the point of tangency. Since you've correctly found the slope at \( x = 2 \) to be \( 19 \), the equation becomes: \[ y - 6 = 19(x - 2) \] Simplifying, you get: \[ y = 19x - 38 + 6 \] \[ y = 19x - 32 \] Now you have the equation of the tangent line! If you want to explore more about differentiating polynomials, you can look into the different rules like the product rule and chain rule. They come in handy for more complex functions and make finding derivatives a breeze! Engage with videos and interactive quizzes online to practice these techniques; they can turn learning into a fun experience. Additionally, once you understand tangent lines, think about their applications in physics, particularly in understanding velocity and acceleration. The concept of tangents can help you predict future positions of a moving object based on its current velocity—it's like having a crystal ball for motion! Exploring calculus through real-world scenarios can spark curiosity and deepen your appreciation for the subject.