Question:
Differentiate w.r.t. \(x\) the function: \((5x)^{3cos2x}\)
Differentiate w.r.t. \(x\) the function: \((5x)^{3cos2x}\)
Updated On: Oct 19, 2023
Hide Solution
Verified By Collegedunia
Solution and Explanation
The correct answer is \((5x)^{3cos2x}[\frac{3cos\,2x}{x}-6sin\,2x\,log\,5x]\)
Let \(y=(5x)^{3cos2x}\)
Taking logarithm on both the sides, we obtain
\(logy=3cos\,2x\,log\,5x\)
Differentiating both sides with respect to \(x\), we obtain
\(\frac{1}{y}\frac{dy}{dx}=3[log5x.\frac{d}{dx}(cos2x)+cos2x\frac{d}{dx}(log5x)]\)
\(⇒\frac{dy}{dx}=3y[log5x(-sin2x).\frac{d}{dx}(2x)+cos2x.\frac{1}{5x}.\frac{d}{dx}(5x)]\)
\(⇒\frac{dy}{dx}=3y[-2sin2xlog5x+\frac{cos2x}{x}]\)
\(⇒\frac{dy}{dx}=3y[\frac{3cos\,2x}{x}-6sin\,2x\,log\,5x]\)
\(∴\frac{dy}{dx}=(5x)^{3cos2x}[\frac{3cos\,2x}{x}-6sin\,2x\,log\,5x]\)
Let \(y=(5x)^{3cos2x}\)
Taking logarithm on both the sides, we obtain
\(logy=3cos\,2x\,log\,5x\)
Differentiating both sides with respect to \(x\), we obtain
\(\frac{1}{y}\frac{dy}{dx}=3[log5x.\frac{d}{dx}(cos2x)+cos2x\frac{d}{dx}(log5x)]\)
\(⇒\frac{dy}{dx}=3y[log5x(-sin2x).\frac{d}{dx}(2x)+cos2x.\frac{1}{5x}.\frac{d}{dx}(5x)]\)
\(⇒\frac{dy}{dx}=3y[-2sin2xlog5x+\frac{cos2x}{x}]\)
\(⇒\frac{dy}{dx}=3y[\frac{3cos\,2x}{x}-6sin\,2x\,log\,5x]\)
\(∴\frac{dy}{dx}=(5x)^{3cos2x}[\frac{3cos\,2x}{x}-6sin\,2x\,log\,5x]\)
Was this answer helpful?
0
0
Top Questions on Continuity and differentiability
- Let $f: [-1, 2] \rightarrow \mathbb{R}$ be given by $f(x) = 2x^2 + x + [x^2] - [x]$, where $[t]$ denotes the greatest integer less than or equal to $t$. The number of points, where $f$ is not continuous, is:
- JEE Main - 2024
- Mathematics
- Continuity and differentiability
- Let \( f : \mathbb{R} \to \mathbb{R} \) be a function given by
\[f(x) = \begin{cases} \frac{1 - \cos 2x}{x^2}, & x < 0 \\\alpha, & x = 0, \text{ where } \alpha, \beta \in \mathbb{R}. \\\beta \sqrt{1 - \cos x} / x, & x > 0 \end{cases} \]
If \( f \) is continuous at \( x = 0 \), then \( \alpha^2 + \beta^2 \) is equal to:- JEE Main - 2024
- Mathematics
- Continuity and differentiability
- For \(a, b > 0\), let \[ f(x) = \begin{cases} \frac{\tan((a+1)x) + b \tan x}{x}, & x < 0, \\ \frac{x}{3}, & x = 0, \\ \frac{\sqrt{ax + b^2x^2} - \sqrt{ax}}{b\sqrt{a x \sqrt{x}}}, & x > 0 \end{cases} \] be a continuous function at \(x = 0\). Then \(\frac{b}{a}\) is equal to:
- JEE Main - 2024
- Mathematics
- Continuity and differentiability
- If the function \( f(x) = \begin{cases} \frac{72^x - 9^x - 8^x + 1}{\sqrt{2} - \sqrt{1 + \cos x}}, & x \neq 0 \\ a \log_e 2 \log_e 3, & x = 0 \end{cases} \) is continuous at \( x = 0 \), then the value of \( a^2 \) is equal to
- JEE Main - 2024
- Mathematics
- Continuity and differentiability
- Let \( f: (0, \pi) \to \mathbb{R} \) be a function given by
\[ f(x) = \begin{cases} \left(\frac{8}{7}\right)^{\tan 8x / \tan 7x}, & 0 < x < \frac{\pi}{2} \\ a - 8, & x = \frac{\pi}{2} \\ \left(1 + |\cot x|\right)^{b^{\lfloor \tan x \rfloor}}, & \frac{\pi}{2} < x < \pi \end{cases} \]
Where \( a, b \in \mathbb{Z} \). If \( f \) is continuous at \( x = \frac{\pi}{2} \), then \( a^2 + b^2 \) is equal to __________.- JEE Main - 2024
- Mathematics
- Continuity and differentiability
View More Questions
Questions Asked in CBSE CLASS XII exam
- Find the inverse of each of the matrices, if it exists. \(\begin{bmatrix} 1 & 3\\ 2 & 7\end{bmatrix}\)
- For what values of x,\(\begin{bmatrix} 1 & 2 & 1 \end{bmatrix}\)\(\begin{bmatrix} 1 & 2 & 0\\ 2 & 0 & 1 \\1&0&2 \end{bmatrix}\)\(\begin{bmatrix} 0 \\2\\x\end{bmatrix}\)=O?
- Find the inverse of each of the matrices,if it exists \(\begin{bmatrix} 2 & 1 \\ 7 & 4 \end{bmatrix}\)
What is the Planning Process?
- CBSE CLASS XII - 2023
- Planning process steps
- Find the inverse of each of the matrices,if it exists. \(\begin{bmatrix} 2 & 3\\ 5 & 7 \end{bmatrix}\)
View More Questions