Page:Calculus Made Easy.pdf/94

 (8) Differentiate $$y^5$$ with respect to $$y^2$$.

(9) Differentiate $$y=\dfrac{\sqrt{1-\theta^2}}{1-\theta}$$.

The process can be extended to three or more differential coefficients, so that $$\dfrac{dy}{dx}=\dfrac{dy}{dz}\times \dfrac{dz}{dv}\times{\dfrac{dv}{dx}}$$

Examples.

(1) If $$z = 3x^4$$; $$v=\dfrac{7}{z^2}$$; $$y=\sqrt{1+v}$$, find $$\dfrac{dv}{dx}$$.

We have

(2) If $$t=\dfrac{1}{5\sqrt{\theta}}$$; $$x= t^3+\dfrac{t}{2}$$; $$v=\dfrac{7x^2}{\sqrt[3]{x-1}}$$, find $$\dfrac{d\phi}{dx}$$.

an expression in which $$x$$ must be replaced by its value, and $$t$$ by its value in terms of $$\theta$$.

(3) If $$\theta=\dfrac{3a^2x}{\sqrt{x^3}}$$; $$\omega=\dfrac{\sqrt{1-\theta^2}}{1+\theta}$$; and $$\phi=\sqrt{3}-\dfrac{1}{\omega\sqrt{2}}$$, find $$\dfrac{d\phi}{dx}$$.