Question
Diffrentiate the following w.r.t.x
$\log \left[\frac{a^{\cos x}}{\left(x^2-3\right)^3 \log x}\right]$
✓
Answer
$\begin{gathered}\quad \text { Let } y=\log \left[\frac{a^{\cos x}}{\left(x^2-3\right)^3 \log x}\right] \\ =\log a^{\cos x}-\log \left(x^2-3\right)^3-\log (\log x) \\ =(\cos x)(\log a)-3 \log \left(x^2-3\right)-\log (\log x)\end{gathered}$
Differentiating w.r.t. $x$, we get
$\frac{d y}{d x}=\frac{d}{d x}\left[(\cos x)(\log a)-3 \log \left(x^2-3\right)-\log (\log x)\right]$
$\begin{aligned} & =(\log a) \cdot \frac{d}{d x}(\cos x)-3 \frac{d}{d x}\left[\log \left(x^2-3\right)\right]-\frac{d}{d x}[\log (\log x)] \\ & =(\log a)(-\sin x)-3 \times \frac{1}{x^2-3} \cdot \frac{d}{d x}\left(x^2-3\right)-\frac{1}{\log x} \cdot \frac{d}{d x}(\log x) \\ & =-(\sin x)(\log a)-\frac{3}{x^2-3} \times(2 x-0)-\frac{1}{\log x} \times \frac{1}{x} \\ & =-(\sin x)(\log a)-\frac{6 x}{x^2-3}-\frac{1}{x \log x} .\end{aligned}$
Need a full question paper?
Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.
Explore more
Similar questions
Reduce each of the following differential equations to the variable separable form and hence solve:
→$(x-y)^2 \frac{d y}{d x}=a^2$
Differentiate the following w.r.t. x:
→$\sin x^x$
Evaluate the following :
→$\int \frac{1}{\sqrt{3 x^2+5 x+7}} \cdot d x$
Compute $\text{P}\Big(\frac{\text{A}}{\text{B}}\Big),$ if P(B) = 0.5 and $\text{P}(\text{A}\cap\text{B})=0.32$
→If $A=\left[\begin{array}{ccc}0 & 4 & 3 \\ 1 & -3 & -3 \\ -1 & 4 & 4\end{array}\right]$, then find $A^2$ and hence find $A^{-1}$
→In the following switching circuit,: Construct switching table
Evaluate the following:
→$\int x^2 \sin 3 x d x$
Represent the following families of curves by forming the corresponding differential equation:
$\text{y}=4\text{ax}$
→$\text{y}=4\text{ax}$
If the percentage error in the radius of a sphere is $\alpha,$ find the percentage error in its volume.
→Evaluate: $\int_{-4}^2 \frac{1}{x^2+4 x+13} d x$
→