Derivatives with Chain Rule

.... area under a curve

... rate of change at a particular value

... gradient of a function

... most optimal solution for a function

... perpendicular gradient of a straight line

8x/(9 - 4x²)^-2

8x/(9 - 4x²)²

-8x/(9 - 4x²)^-2

-8x(9 - 4x²)²

$$\frac{\text{d}y}{\text{d}x}=\frac{3x+1}{3x^2+x}$$  

$$\frac{\text{d}y}{\text{d}x}=\frac{6x+1}{3x^2+x}$$  

$$\frac{\text{d}y}{\text{d}x}=\frac{6x}{3x^2+x}$$  

$$\frac{\text{d}y}{\text{d}x}=\frac{1}{3x^2+x}$$  

$$\frac{\text{d}y}{\text{d}x}=\frac{10}{5x^2}$$  

$$\frac{\text{d}y}{\text{d}x}=\frac{5}{x^2}$$  

$$\frac{\text{d}y}{\text{d}x}=\frac{2}{x}$$  

$$\frac{\text{d}y}{\text{d}x}=\frac{2}{x^2}$$  

$$\frac{\text{d}y}{\text{d}x}=\frac{1}{2\left(3x+5\right)}$$  

$$\frac{\text{d}y}{\text{d}x}=\sqrt[]{\frac{3}{\left(3x+5\right)}}$$  

$$\frac{\text{d}y}{\text{d}x}=\frac{3}{\left(3x+5\right)}$$  

$$\frac{\text{d}y}{\text{d}x}=\frac{3}{2\left(3x+5\right)}$$  

$$\frac{\text{d}y}{\text{d}x}=2xe^{3-x^2}$$  

$$\frac{\text{d}y}{\text{d}x}=-2xe^{3-x^2}$$  

$$\frac{\text{d}y}{\text{d}x}=\left(3-2x\right)e^{3-x^2}$$  

$$\frac{\text{d}y}{\text{d}x}=-x^2e^{3-x^2}$$