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Question

If $f left parenthesis x comma y right parenthesis equals S i n invisible function application open parentheses e to the power of a x end exponent plus e to the power of b y end exponent close parentheses$ then $f subscript x y end subscript equals$

$a b e^{a x + b y} \cdot S i n (e^{a x} + e^{b y})$
$- a b e^{a x + b y} S i n (e^{a x} + e^{b y})$
$a b e^{a x + b y} C o s (e^{a x} + e^{b y})$
$- a b e^{a x + b y} C o s (e^{a x} + e^{b y})$

Hint:

We are given a function. It is function of two variables x and y. We should to find the value of f_xy.

The correct answer is: $negative a b e to the power of a x plus b y end exponent S i n invisible function application open parentheses e to the power of a x end exponent plus e to the power of b y end exponent close parentheses$

The given function is f(x,y) = sin(e^ax+ e^by)
We have to find the value of f_xy
It means we have to find the value of $fraction numerator partial differential squared f over denominator partial differential x partial differential y end fraction$
We will first take partial derivative w.r.t to one of the variable. Then we take the partial derivative of that value w.r.t the remaining variable.
Taking the partial derivative w.r.t x
$f open parentheses x comma y close parentheses equals sin open parentheses e to the power of a x end exponent plus e to the power of b y end exponent close parentheses fraction numerator partial differential f over denominator partial differential x end fraction equals cos left parenthesis e to the power of a x end exponent plus e to the power of b y end exponent right parenthesis space fraction numerator partial differential over denominator partial differential x end fraction left parenthesis e to the power of a x end exponent plus e to the power of b y end exponent right parenthesis space space space space space space space space space equals cos left parenthesis e to the power of a x end exponent plus e to the power of b y end exponent right parenthesis left parenthesis e to the power of a x end exponent plus 0 right parenthesis fraction numerator partial differential over denominator partial differential x end fraction a x space space space space space space space space space equals a e to the power of a x end exponent cos left parenthesis e to the power of a x end exponent plus e to the power of b y end exponent right parenthesis$
We will take the derivative of this value w.r.t y
$fraction numerator partial differential f over denominator partial differential x end fraction equals a e to the power of a x end exponent cos left parenthesis e to the power of a x end exponent plus e to the power of b y end exponent right parenthesis fraction numerator partial differential squared f over denominator partial differential x partial differential y end fraction space equals a e to the power of a x end exponent fraction numerator partial differential over denominator partial differential x end fraction cos left parenthesis e to the power of a x end exponent plus e to the power of b y end exponent right parenthesis space space space space space space space space space space space space space space space equals a e to the power of a x end exponent left square bracket negative sin left parenthesis e to the power of a x end exponent plus e to the power of b x end exponent right parenthesis right square bracket fraction numerator partial differential over denominator partial differential x end fraction e to the power of a x end exponent plus e to the power of b y space end exponent space space space space space space space space space space space space space space space equals negative a e to the power of a x end exponent sin left parenthesis e to the power of a x end exponent plus e to the power of b y end exponent right parenthesis left parenthesis 0 plus e to the power of b y end exponent right parenthesis fraction numerator partial differential over denominator partial differential x end fraction b y space space space space space space space space space... left curly bracket fraction numerator d over denominator d x end fraction e to the power of f left parenthesis x right parenthesis end exponent equals e to the power of f left parenthesis x right parenthesis end exponent fraction numerator d f left parenthesis x right parenthesis over denominator d x end fraction right curly bracket space space space space space space space space space space space space space space space equals negative a e to the power of a x end exponent sin left parenthesis e to the power of a x end exponent plus e to the power of b y end exponent right parenthesis left parenthesis e to the power of b y end exponent right parenthesis b space space space space space space space space space space space space f subscript x y end subscript space space equals negative a b e to the power of a x plus b y end exponent sin left parenthesis e to the power of a x end exponent plus e to the power of b y end exponent right parenthesis space space space space space space space space space space space space space space space space... left curly bracket e to the power of m e to the power of n equals e to the power of m plus n end exponent right curly bracket space space space space space space space space space$
This is the required answer.

We have used rules of indices and different formulae of differentiation to solve the question. When we find the derivative w.r.t to some variable we take another variable as consta.

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We are given a function. It is function of two variables x and y. We should to find the value of fxy.

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If $f left parenthesis x comma y right parenthesis equals S i n invisible function application open parentheses e to the power of a x end exponent plus e to the power of b y end exponent close parentheses$ then $f subscript x y end subscript equals$

We are given a function. It is function of two variables x and y. We should to find the value of f_xy.

The correct answer is: $negative a b e to the power of a x plus b y end exponent S i n invisible function application open parentheses e to the power of a x end exponent plus e to the power of b y end exponent close parentheses$

The radius of the earth is 6400 km and g= $10 space m s squared$ . In order that a body of 5 kg weights zero at the equator, the angular speed of the earth is =……rad/sec

The radius of the earth is 6400 km and g= $10 space m s squared$ . In order that a body of 5 kg weights zero at the equator, the angular speed of the earth is =……rad/sec

A transverse wave is represented by $y equals A S i n left parenthesis u t minus k x right parenthesis$ . For what value of its wavelength will the wave velocity be equal to the maximum velocity of the particle taking part in the wave propagation?

A transverse wave is represented by $y equals A S i n left parenthesis u t minus k x right parenthesis$ . For what value of its wavelength will the wave velocity be equal to the maximum velocity of the particle taking part in the wave propagation?

If. two almost identical waves having frequencies $n subscript 1 end subscript$ and $n subscript 2 end subscript$ produced one after the other superposes then the time interval to obtain a beat of maxirmum intensity is .......

If. two almost identical waves having frequencies $n subscript 1 end subscript$ and $n subscript 2 end subscript$ produced one after the other superposes then the time interval to obtain a beat of maxirmum intensity is .......

As shown in figure, two pulses in a string having center to center distance of 8 cm are travelling abng mutually opposite direction. If the speed of both the pulse is $2 text end text c m divided by$ s, then after 2s, the energy of these pulses will be…………

As shown in figure, two pulses in a string having center to center distance of 8 cm are travelling abng mutually opposite direction. If the speed of both the pulse is $2 text end text c m divided by$ s, then after 2s, the energy of these pulses will be…………

The gravitational force between two point masses $m subscript 1$ and $m subscript 2$ at separation r is given by F= $G fraction numerator m subscript 1 m subscript 2 over denominator r squared end fraction$ The constant K…..

The gravitational force between two point masses $m subscript 1$ and $m subscript 2$ at separation r is given by F= $G fraction numerator m subscript 1 m subscript 2 over denominator r squared end fraction$ The constant K…..

Three equal masses of mkg each are plced the vertices of an equilateral triangle PQR and a mass of 2 m kg is placed at the centroid 0 of the triangle which is at a distance of $\sqrt{2}$ m from each of vertices of triangle. The force in newton. acting on the mass 2 m is …….

Three equal masses of mkg each are plced the vertices of an equilateral triangle PQR and a mass of 2 m kg is placed at the centroid 0 of the triangle which is at a distance of $\sqrt{2}$ m from each of vertices of triangle. The force in newton. acting on the mass 2 m is …….

The displacement of a particle is given by $x equals A$ $C o s invisible function application omega t.$ Which of the following graph represents variation in potential energy as a fimction of tims $t$ and displacement $x$ .

The displacement of a particle is given by $x equals A$ $C o s invisible function application omega t.$ Which of the following graph represents variation in potential energy as a fimction of tims $t$ and displacement $x$ .

Two blocks $A$ and $B$ are attached to the two ends of a spring having leagth $L$ and force consiant $k$ on a horizontal surface. Initially the systen is in equilhium. Now a third block having same mass $m$ , moving with velocity $v$ collides with block A. Ln this situation........

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