Page:Calculus Made Easy.pdf/50

 Some Further Examples.

The following further examples, fully worked out, will enable you to master completely the process of differentiation as applied to ordinary algebraical expressions, and enable you to work out by yourself the examples given at the end of this chapter.

(1) Differentiate $$y=\frac{x^5}{7}-\frac{3}{5}$$.

$$\frac {3}{5}$$ is an added constant and vanishes (see p. 26).

We may then write at once $\frac {dy}{dx}=\frac {1}{7}\times 5\times x^{5-1}$, or $\frac{dy}{dx}=\frac {5}{7}x^4$.

(2) Differentiate $$y=a\sqrt x-\frac{1}{2}\sqrt a$$.

The term $$\frac{1}{2}\sqrt a$$ vanishes, being an added constant; and as $$a\sqrt x$$, in the index form, is written $$ax^{\frac {1}{2}}$$, we have

$\frac{dy}{dx}=a\times \frac{1}{2}\times x^{\frac{1}{2}-1}=\frac {a}{2}\times x^{-\frac {1}{2}}$,|undefined

or

$\frac{dy}{dx}=\frac {a}{2\sqrt x}$.

(3) If $$ay+bx=by-ax+(x+y)\sqrt {a^2-b^2}$$, find the differential coefficient of $$y$$ with respect to $$x$$.

As a rule an expression of this kind will need a little more knowledge than we have acquired so far;