Answer to Question #185294 in Differential Equations for Gatare james

Question

Differentiate the following and calculate the value of 𝒅𝒚 at the value of x stated: 𝒅𝐱

1. 𝒚=𝟐𝒙𝟑 +𝟒𝒙𝟐 −𝟐𝒙+𝟕 at 𝒙=−𝟐

2. 𝒚=𝟑𝒙𝟒 −𝟓𝒙𝟑 +𝟒𝒙𝟐 −𝒙+𝟒 at 𝒙=𝟑

Answer 1

1. $𝒚=𝟐𝒙^𝟑 +𝟒𝒙^𝟐 −𝟐𝒙+𝟕\\$

Differentiate this equation with respect to x

$\dfrac{dy}{dx}=6x^2+8x-2$

So $\dfrac{dy}{dx}$ at $x=-2$ so put the value of $x=-2$ in above equation

$\Rightarrow 6\times(-2)^2+8(-2)-2=6$

2.$𝒚=𝟑𝒙^𝟒 −𝟓𝒙^𝟑 +𝟒𝒙^𝟐 −𝒙+𝟒$

Differentiate this equation with respect to x

$\dfrac{dy}{dx}=12x^3-15x^2+8x-1$

So $\dfrac{dy}{dx}$ at $x=3$ so put the value of $x=3$ in above equation

$\Rightarrow 12\times(3)^3-15\times(3)^2+8\times(3)-1$ $=212$