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Find $\dfrac{d}{dx}\,\cos{\big(\log_e{(x)}\big)}$ by Chain rule

The cosine of natural logarithm of $x$ is a composite function. Its derivative cannot be calculated by either derivative rule of cosine and differentiation law of logarithm. However, they can be used by chain rule to find the differentiation of cos of neper logarithm of $x$ with respect to $x$.

$\dfrac{d}{dx}\,\cos{\big(\log_e{(x)}\big)}$

Let’s learn how to evaluate the derivative of cosine of natural logarithm of $x$ with respect to $x$ by the chain rule in fundamental notation.

Compare the function with Chain Rule

A composite function is usually denoted by by $f\Big(g(x)\Big)$ in the fundamental representation of the chain rule. In this problem, the composite function is $\cos{\big(\log_e{(x)}\big)}$. Let’s compare both functions firstly before finding the derivative of the composition of the functions by chain rule.

Suppose, $f\Big(g(x)\Big)$ $\,=\,$ $\cos{\big(\log_e{(x)}\big)}$, then $g(x) \,=\, \log_e{(x)}$

The chain rule in the fundamental notation is written as follows.

$\dfrac{d}{dx}f\Big(g(x)\Big)$ $\,=\,$ $\dfrac{d}{d\,g(x)}f\Big(g(x)\Big) \times \dfrac{d}{dx}\,g(x)$

Now, substitute the functions in the fundamental notation of the chain rule.

$\,\,\,\therefore\,\,\,\,\,\,$ $\dfrac{d}{dx}\,\cos{\big(\log_e{(x)}\big)}$ $\,=\,$ $\dfrac{d}{d\,\log_e{(x)}}\,\cos{\big(\log_e{(x)}\big)}$ $\times$ $\dfrac{d}{dx}\,\log_e{(x)}$

Find Derivative with respect to internal function

According to the chain rule, the differentiation of the cosine of natural logarithm of $x$ is written as a product of the derivative of the cosine of natural logarithm of $x$ with respect to natural logarithm of $x$ and the derivative of natural logarithm of $x$ with respect to $x$.

$\dfrac{d}{d\,\log_e{(x)}}\,\cos{\big(\log_e{(x)}\big)}$ $\times$ $\dfrac{d}{dx}\,\log_e{(x)}$

Let us focus on the first factor. For avoiding confusion, take $y \,=\, \log_e{x}$ but keep the second factor as it is.

$=\,\,\,$ $\dfrac{d}{dy}\,\cos{(y)}$ $\times$ $\dfrac{d}{dx}\,\log_e{(x)}$

$=\,\,\,$ $\dfrac{d}{dy}\,\cos{y}$ $\times$ $\dfrac{d}{dx}\,\log_e{(x)}$

The derivative of cosine of $y$ with respect to $y$ is equal to negative sine of angle $y$ as per the derivative rule of cosine function.

$=\,\,\,$ $-\sin{y}$ $\times$ $\dfrac{d}{dx}\,\log_e{(x)}$

Now, replace the value of variable $y$ by its actual value.

$=\,\,\,$ $-\sin{\big(\log_e{(x)}\big)}$ $\times$ $\dfrac{d}{dx}\,\log_e{(x)}$

$=\,\,\,$ $-\sin{\big(\log_e{x}\big)}$ $\times$ $\dfrac{d}{dx}\,\log_e{(x)}$

Find the differentiation of the Internal function

It is time to concentrate on the second factor. The derivative of the natural logarithm of $x$ with respect to $x$ can be calculated as per the derivative rule of logarithm.

$=\,\,\,$ $-\sin{\big(\log_e{x}\big)}$ $\times$ $\dfrac{1}{x}$

$=\,\,\,$ $-\dfrac{\sin{\big(\log_e{x}\big)} \times 1}{x}$

$=\,\,\,$ $-\dfrac{\sin{\big(\log_e{x}\big)}}{x}$

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