theta = arctan(5x/2). Using implicit differentiation, find d theta/dx.

First, we must rearrange to give 2tan(θ) = 5x. Differentiate both sides with respect to x: 2sec²(θ)dθ/dx = 5 Use identity sin²(θ) + cos²(θ) = 1, dividing through by cos²(θ), to get 2(1+tan²(θ))dθ/dx=5. From earlier, we know that tan(θ) = 5x/2, so substituting gives 2(1+25x²/4) dθ/dx= 5 dθ/dx = 5/(2+25x²/2)