Intereting Posts

Prove this inequality with $xyz\le 1$
TVS: Topology vs. Scalar Product
Existence of additive transformation of random variables
Limit of $L^p$ norm
How to find the radius of convergence?
sum of the product of consecutive legendre symbols is -1
Calculate coordinates of 3rd point (vertex) of a scalene triangle if angles and sides are known.
Intuition behind conjugation in group theory
How prove this $\prod_{1\le i<j\le n}\frac{a_{j}-a_{i}}{j-i}$ is integer
How to determine the existence of all subsets of a set?
A proof of the Lagrange's theorem on cyclic extention fields
A UFD for which the related formal power series ring is not a UFD
Mean value theorem application for multivariable functions
Finding real roots of $ P(x)=x^8 – x^7 +x^2 -x +15$
prove that the greatest number of regions that $n \geq 1$ circles can divide the plane is $n^2-n+2$

How can I differentiate $\displaystyle \sin{x}^{\cos{x}}$? I know the power rule will not work in this case, but logarithmic differentiation would. I’m not sure how to start the problem though and I’m not too comfortable with logarithmic differentiation.

- How do I evaluate this integral $\int_0^\pi{\frac{{{x^2}}}{{\sqrt 5-2\cos x}}}\operatorname d\!x$?
- An inflection point where the second derivative doesn't exist?
- $\int_0^1 {\frac{{\ln (1 - x)}}{x}}$ without power series
- Showing $\lim_{n\rightarrow\infty}\sqrt{n^3+n^2}-\sqrt{n^3+1}\rightarrow\frac{1}{3}$
- How to find the closed form of the integral$\int_{0}^{\infty}f(x)\frac{\sin^nx}{x^m}dx$
- Probability that a random permutation has no fixed point among the first $k$ elements
- Which function ($f$) is continuous nowhere but $|f(x)|$ is continuous everywhere?
- A fractional part integral giving $\frac{F_{n-1}}{F_n}-\frac{(-1)^n}{F_n^2}\ln\left(\!\frac{F_{n+2}-F_n\gamma}{F_{n+1}-F_n\gamma}\right)$
- Prove $\int_a^b f(x)dx \leq \frac{e^{2L\beta}-1}{2L\alpha}\int_c^d f(x)dx$
- Derivative of cross-product of two vectors

Let $y = \sin x ^{\cos x } $ , then

$$ \ln y = \cos x \ln (\sin x ) \implies \frac{y’}{y} = -\sin x \ln(\sin x) + \cos x\frac{\cos x}{\sin x}$$

$$ \therefore y’ = \sin x ^{\cos x } \left( \frac{\cos^2 x}{\sin x} – \sin x \ln(\sin x)\right) $$

In general,

Let $f, g$ be any functions. Let $y = f^g \implies \ln y = g \ln f $

$$ \therefore \frac{y’}{y} = g’ \ln f + g\frac{f’}{f} \implies \frac{ df^g}{dx}= y’ = f^g \left( f’ \ln f + \frac{g f’}{f} \right)$$

In addition to the other answers, you may also rewrite the function as

$$[e^{\ln(\sin x)}]^{\cos x}=e^{\ln(\sin x)\cos x}$$

which you should be able to take the derivative of through the chain rule and product rule.

Let $f(x) = sinx^{cosx}$ then $log(f(x)) = cosx log sinx $. Now differebtiate LHS by function of function rule and RHS bu product rule and finally solve for $f(x)$.

Consider $$y=\sin(x)^{\cos(x)}.$$ We will use the method of logarithmic differentiation to obtain this functions derivative. Take the natural logarithm of both sides of the equation and use the properties of logarithms to simplify. So $$\ln(y)=\cos(x)\cdot \ln(\sin(x)).$$ Differentiating implicitly with respect to $x$ we obtain $${1\over y}\cdot y’=\cos(x)\cdot {\cos(x)\over \sin(x)}-\sin(x)\cdot \ln(\sin(x)).$$ Simplifying the right-hand side we see that $${1\over y}\cdot y’=\cos(x)\cdot \cot(x)-\sin(x)\cdot \ln(sin(x)).$$ Multiply both sides of the equation by $y$ and we have $$y’=\sin(x)^{\cos(x)}(\cos(x)\cdot \cot(x)-\sin(x)\cdot \ln(\sin(x)).$$

- The automorphism group of the real line with standard topology
- the product of diagonals of a regular polygon
- What does insolvability of the quintic mean exactly?
- Proving $\left(A-1+\frac1B\right)\left(B-1+\frac1C\right)\left(C-1+\frac1A\right)\leq1$
- Prove every odd integer is the difference of two squares
- Can an odd perfect number be divisible by $101$?
- Proving an identity involving factorials
- Why can't epsilon depend on delta instead?
- Prove that $\lambda = 0$ is an eigenvalue if and only if A is singular.
- Do there exist bijections between the following sets?
- How could we define the factorial of a matrix?
- Is $\int_{\mathbb R} f(\sum_{k=1}^n\frac{1}{x-x_k})dx$ independent of $x_k$'s for certain $f$?
- How to geometrically show that there are $3$ $D_4$ subgroups in $S_4$?
- Properties of matrices changing with the parity of matrix dimension
- Explanation of Lagrange Interpolating Polynomial