Elements of the Differential and Integral Calculus/Chapter V part 2

44. Differentiation of a logarithm.

Let $$y = \log_a v$$.

Differentiating by the General Rule, p. 29 [§ 31], considering $$v$$ as the independent variable, we have

Since $$v$$ is a function of $$x$$ and it is required to differentiate $$\log_a v$$ with respect to $$x$$, we must use formula (A), § 42, for differentiating a function of a function, namely,

The derivative of the logarithm of a function is equal to the product of the modulus of the system of logarithms and the derivative of the function, divided by the function.

45. Differentiation of the simple exponential function.

The derivative of a constant with a variable exponent is equal to the product of the natural logarithm of the constant, the constant with the variable exponent, and the derivative of the exponent.

46. Differentiation of the general exponential function.

The derivative of a function with a variable exponent is equal to the sum of the two results obtained by first differentiating by VI', ''regarding the exponent as constant,. and again differentiating by IX, regarding the function as constant.''

Let v = n, any constant; then X reduces to


 * $$\frac{d}{dx}(u^n) = nu^{n - 1} \frac{du}{dx}$$.

But this is the form differentiated in § 40; therefore VI holds true for any value of $$n$$.

47. Logarithmic differentiation. Instead of applying VIII and VIIIa at once in differentiating logarithmic functions, we may sometimes simplify the work by first making use of one of the formulas 7-10 on p. 1 [§ 1]. Thus above Illustrative Example 2 may be solved as follows:

In differentiating an exponential function, especially a variable with a variable exponent, the best plan is first to take the logarithm of the function and then differentiate. Thus Illustrative Example 5, p. 50 [§ 46], is solved more elegantly as follows:

ILLUSTRATIVE EXAMPLE 4. Differentiate $$y = (4x^2 - 7)^{2 + \sqrt{x^2 - 5}}$$.

Solution. Taking the logarithm of both sides,


 * $$\log y = (2 + \sqrt{x^2 - 5}) \log (4x^2 - 7)$$.

Differentiating both sides with respect to $$x$$,




 * style="text-align: right;"|$$\frac{1}{y} \frac{dy}{dx}$$
 * $$= (2 + \sqrt{x^2 - 5}) \frac{8x}{4x^2 - 7} + \log(4x^2 - 7) \cdot \frac{x}{\sqrt{x^2 - 5}}$$.
 * style="text-align: right;"|$$\frac{dy}{dx}$$
 * $$= x(4x^2 - 7)^{2 + \sqrt{x^2 - 5}} \left [ \frac{8(2 + \sqrt{x^2 - 5})}{4x^2 - 7} + \frac{\log (4x^2 - 7)}{\sqrt{x^2 - 5}} \right ]$$. Ans.
 * }
 * }

In the case of a function consisting of a number of factors it is sometimes convenient to take the logarithm before differentiating. Thus,

ILLUSTRATIVE EXAMPLE 5. Differentiate $$y = \sqrt{\frac{(x - 1)(x - 2)}{(x - 3)(x - 4)}}$$.

Solution. Taking the logarithm of both sides,


 * $$\log y = \frac{1}{2} [\log (x -1) + \log (x - 2) - \log(x - 3) - \log(x - 4)]$$.

Differentiating both sides with respect to $$x$$,

EXAMPLES

Differentiate the following:

12. $$\frac{d}{dx} e^{ax} = ae^{ax}$$.

13. $$\frac{d}{dx} e^{4x + 5} = 4e^{4x + 5}$$.

14. $$\frac{d}{dx} a^{3x} = 3a^{3x} \log a$$.

15. $$\frac{d}{dt} \log(3 - 2t^2) = \frac{4t}{2t^2 - 3}$$.

16. $$\frac{d}{dy} \log \frac{1 + y}{1 - y} = \frac{2}{1 - y^2}$$.

17. $$\frac{d}{dx}e^{b^2 + x^2} = 2xe^{b^2 + x^2}$$.

18. $$\frac{d}{d\theta} a^{\log a} = \frac{1}{\theta} a^{\log \theta} \log a$$.

19. $$\frac{d}{ds}b^{s^2} = 2x \log b \cdot b^{s^2}$$.

20. $$\frac{d}{dv} ae^{\sqrt{v}} = \frac{ae^{\sqrt{v}}}{2\sqrt{v}}$$.

21. $$\frac{d}{dx} a^{e^x} = \log a \cdot a^{e^x} \cdot e^x$$.

25. $$\frac{d}{dx} \left [ e^x ( 1- x^2 \right ] = e^x (1 - 2x - x^2)$$.

26. $$\frac{d}{dx} \left ( \frac{e^x - 1}{e^x + 1} \right ) = \frac{2e^x}{(e^x + 1)^2}$$

27. $$\frac{d}{dx} \left ( x^2 e^{ax} \right ) = xe^{ax}(ax + 2)$$.

50. Differentiate the following functions:

48. Differentiation of $$\sin v$$.

[ Since $$ \lim_{\Delta v \to 0} \left ( \frac{\sin \frac{\Delta v}{2}}{\frac{\Delta v}{2}} \right ) = 1,$$ by § 22, p. 21, and $$\lim_{\Delta v \to 0} \cos \left ( v + \frac{\Delta v}{2} \right ) = \cos v$$ ].

Since $$v$$ is a function of $$x$$ and it is required to differentiate $$\sin v$$ with respect to $$x$$, we must use formula (A), § 42, for differentiating a function of a function, namely,

The statement of the corresponding rules will now be left to the student.

49. Differentiation of $$\cos v$$.

50. Differentiation of $$\tan v$$.

51. Differentiation of $$\cot v$$.

52. Differentiation of $$\sec v$$.

53. Differentiation of $$\csc v$$.