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Open in new tab Formulas: Derivatives of logarithmic functions.
![\[\begin{aligned} & \text { (1) } \frac{d}{d x}[\ln x]=\frac{1}{x}, \quad x>0 \\ \\ & \text { (2) } \frac{d}{d x}\left[\log _{b}^{x}\right]=\frac{1}{x \ln b}, x>0 \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-510bdd9d06d45d12851159516659c72a_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{aligned} & \text { (1) } \frac{d}{d x}[\ln u]=\frac{1}{u} \frac{d u}{d x} \\ \\ & \text { (2) } \frac{d}{d x}\left[\log _{b} u\right]=\frac{1}{u \ln b} \frac{d u}{d x} \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-698dec286a3e6e9628b1519a34024d47_l3.png "Rendered by QuickLaTeX.com")
![\[\text { (3) }\frac{d}{d x}[\ln |x|]=\frac{1}{x} , \text { if } x \neq 0\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-d1c7476a0b654dacc1153614547dbcc1_l3.png "Rendered by QuickLaTeX.com")
Q.01 Find 
where 
.
Solution:
![\[\quad y=\ln 5 x\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-43e135318e7c7ccd40f3e4c64a8d73ab_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{aligned} \frac{d y}{d x} & =\frac{d}{d x}[\ln 5 x] \\ \\ \frac{d y}{d x} & =\frac{1}{5 x} \frac{d(5 x)}{d x} \\ \\ \frac{d y}{d x} & =\frac{1}{5 x}(5)(1)=\frac{5}{5 x} \\ \\ \frac{d y}{d x} & =\frac{1}{x} \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-d53b2b8e375297d9fd06ae177c7186bb_l3.png "Rendered by QuickLaTeX.com")
Q.02 Find where 
.
Solution:
![\[\quad y=\ln (x / 3)\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-ae613f8111205bfc464248bc28271cc8_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{aligned} & \frac{d y}{d x}=\frac{d}{d x}[\ln (x / 3)] \\ \\ & \frac{d y}{d x}=\frac{1}{x / 3} \cdot \frac{d}{d x}(x / 3)=\frac{1}{(x / 3)} \cdot \frac{1}{3}(1) \\ \\ & \frac{d y}{d x}=\frac{1}{3 \cdot x / 3}=\frac{1}{x} \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-0ff9249e13d6ba4bde465e3ae7a36a5d_l3.png "Rendered by QuickLaTeX.com")
Q.03 Find where 
.
Solution:
![\[y=\ln |1+x|\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-c68e3cb0afab55e08f390939d3c5c086_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{aligned} & \text { Differentiating w.r.t } x \\ \\ & \frac{d y}{d x}=\frac{d}{d x}[\ln |1+x|] \\ \\ & \frac{d y}{d x}=\frac{1}{1+x} \frac{d(1+x)}{d x} \\ \\ & \text {because} \frac{d}{d x} \ln |x|=\frac{1}{x} \text { ,if } x \neq 0 \\ \\ & \therefore \frac{d y}{d x}=\frac{1}{1+x}(0+1)=\frac{1}{d x} \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-96637498ac3cd26ab27f53ab0161a90f_l3.png "Rendered by QuickLaTeX.com")
Q.04 Find where 
.
Solution:
![\[y=\ln (2+\sqrt{x}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-443ba5c8c24d44c6e854a5f53cb6843c_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{aligned} & \frac{d y}{d x}=\frac{d}{d x}[\ln (2+\sqrt{x})] \\ \\ & \frac{d y}{d x}=\frac{1}{2+\sqrt{x}} \frac{d}{d x}[2+\sqrt{x}] \\ \\ & \frac{d y}{d x}=\frac{1}{2+\sqrt{x}}\left[0+\frac{d}{d x} x^{1 / 2}\right] \\ \\ & \frac{d y}{d x}=\frac{1}{2+\sqrt{x}}\left[\frac{1}{2} x^{\frac{1}{2}-1}\right] \\ \\ & \frac{d y}{d x}=\frac{1}{2+\sqrt{x}} \cdot \frac{1}{2} x^{-1 / 2} \\ \\ & \frac{d y}{d x}=\frac{1}{2 x^{1 / 2}(2+\sqrt{x})} \\ \\ & \frac{d y}{d x}=\frac{1}{2 \sqrt{x}(2+\sqrt{x})}=\frac{1}{4 \sqrt{x}+2(\sqrt{x})^{2}}=\frac{1}{4 \sqrt{x}+2 x} \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-19d472f9e6899a9d23cfb47da3197d62_l3.png "Rendered by QuickLaTeX.com")
Q.05 Find where 
.
Solution:
![\[y=\ln \left|x^{2}-1\right|\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-2cd7a50d5a08e41303cf5c30adf5a48c_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{aligned} & \frac{d y}{d x}=\frac{d}{d x}\left[\ln \left|x^{2}-1\right|\right] \\ \\ & \frac{d y}{d x}=\frac{1}{x^{2}-1} \frac{d}{d x}\left(x^{2}-1\right) \\ \\ & \frac{d y}{d x}=\frac{1}{x^{2}-1}\left(2 x^{2-1}-0\right)=\frac{2 x}{x^{2}-1} \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-87637f48f3812da8e9771e36f0277352_l3.png "Rendered by QuickLaTeX.com")
Q.06 Find where 
.
Solution:
![\[y=\ln \left|x^{3}-7 x^{2}-3\right|\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-b279b146b0fb6b716047ce6c0bf1c95f_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{aligned} & \frac{d y}{d x}=\frac{d}{d x}\left[\ln \left|x^{3}-7 x^{2}-3\right|]. \\ \\ & \frac{d y}{d x}=\frac{1}{x^{3}-7 x^{2}-3} \cdot \frac{d}{d x}\left[x^{3}-7 x^{2}-3\right] \\ \\ & \frac{d y}{d x}=\frac{1}{x^{3}-7 x^{2}-3} \cdot\left[3 x^{3-1}-7\left(2 x^{2-1}\right)-0\right] \\ \\ & \frac{d y}{d x}=\frac{1}{x^{3}-7 x^{2}-3} \cdot\left[3 x^{2}-14 x\right]=\frac{3 x^{2}-14 x}{x^{3}-7 x^{2}-3} \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-1a66ced2d5358e6bb248d2de235c17c3_l3.png "Rendered by QuickLaTeX.com")
Q.07 Find where 
.
Solution:
![\[y=\ln \left(\frac{x}{x^{2}+1}\right)\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-dbd9a8bf573d64690e4de4915ffb5027_l3.png "Rendered by QuickLaTeX.com")
![\[\begin{aligned} & \frac{d y}{d x}=\frac{d}{d x}\left[\ln \left(\frac{x}{1+x^{2}}\right)\right] \quad \because \ln \left(\frac{x}{y}\right)=\ln x-\ln y \\ \\ & \frac{d y}{d x}=\frac{d}{d x}\left[\ln x-\ln \left(1+x^{2}\right)\right] \\ \\ & \frac{d y}{d x}=\frac{d}{d x}[\ln x]-\frac{d}{d x}\left[\ln \left(1+x^{2}\right)\right] \\ \\ & \frac{d y}{d x}=\frac{1}{x}-\frac{1}{1+x^{2}} \frac{d}{d x}\left(1+x^{2}\right) \\ \\ & \frac{d y}{d x}=\frac{1}{x}-\frac{1}{1+x^{2}}\left(0+2 x^{2-1}\right) \\ \\ & \frac{d y}{d x}=\frac{1}{x}-\frac{2 x}{1+x^{2}} \\ \\ & \frac{d y}{d x}=\frac{1+x^{2}-2 x^{2}}{x\left(1+x^{2}\right)} =\frac{1-x^{2}}{x\left(1+x^{2}\right)} \end{aligned}\]](https://genesismath.com/wp-content/ql-cache/quicklatex.com-d020def6460f0e4ec26dc8ddbf313949_l3.png "Rendered by QuickLaTeX.com")