1. differentiate cos(3/x)
2. differentiate sin(4/x)
3. differentiate 3/{sin(3x+pi)}
4. differentiate pxsin(q/x)where p and q are constants.
5. differentiate xsin(a/x) where a is constant
6. differentiate sec^3(3x^2+1)
To differentiate each of the given functions:
1. To differentiate cos(3/x), we can use the chain rule.
Start by letting u = 3/x, then differentiate the outer function cos(u) with respect to u, which gives -sin(u).
Next, differentiate the inner function u = 3/x with respect to x, which gives du/dx = -3/x^2.
Now, multiply these two differentials together to get the final result: -sin(u) * (-3/x^2) = 3sin(3/x)/x^2.
2. To differentiate sin(4/x), we can again use the chain rule.
Let u = 4/x, then differentiate the outer function sin(u) with respect to u, which gives cos(u).
Differentiating the inner function u = 4/x with respect to x gives du/dx = -4/x^2.
Multiplying these two differentials together yields: cos(u) * (-4/x^2) = -4cos(4/x)/x^2.
3. To differentiate 3/{sin(3x+pi)}, we can use the quotient rule.
First, differentiate the numerator 3 with respect to x, which gives 0.
Next, differentiate the denominator sin(3x+pi) with respect to x, which gives cos(3x+pi) * d(3x+pi)/dx.
Since d(3x+pi)/dx is simply equal to 3, we can rewrite the derivative of the denominator as 3cos(3x+pi).
Now, apply the quotient rule: (0 * sin(3x+pi) - 3 * cos(3x+pi))/ (sin(3x+pi))^2.
Simplifying the expression gives -3cos(3x+pi)/sin^2(3x+pi).
4. To differentiate pxsin(q/x), where p and q are constants, we again use the chain rule.
First, differentiate the outer function pxsin(q/x) with respect to x: this gives p * d(sin(q/x))/dx.
Next, differentiate the inner function q/x with respect to x, which gives dq/dx * (1/x).
Now, differentiate sin(q/x) with respect to its argument q/x: this gives cos(q/x).
Combining these differentials, we have: p * dq/dx * (1/x) * cos(q/x).
5. To differentiate xsin(a/x), where a is a constant, we use the product rule.
Differentiate the first term x with respect to x, which gives 1.
Next, differentiate the second term sin(a/x) with respect to its argument a/x, giving cos(a/x) * (-1/x^2).
Multiply these differentials together to get: 1 * sin(a/x) + x * cos(a/x) * (-1/x^2), which simplifies to sin(a/x) - (x * cos(a/x)/x^2).
6. To differentiate sec^3(3x^2+1), we can use the chain rule.
First, differentiate the outer function sec^3(u) with respect to u, which gives 3sec^2(u) * d(u)/d(x).
Next, find the derivative of the inner function u = 3x^2+1 with respect to x, which gives du/dx = 6x.
Multiply these differentials together to get the final result: 3sec^2(3x^2+1)*6x = 18x * sec^2(3x^2+1).
Remember to always double-check your work and simplify your answers as needed.