What is the derivative of:
f(t) = e^{t} \mathrm{sech}(t)
f(x)=y^3x (natural log)
f(x)=sinh(x)tanh(x)
f(x)=8sin(5x)arcsinx)
To determine the derivative of each of these functions, we can use basic rules of differentiation. I will explain the steps for finding the derivatives of each function.
1. f(t) = e^t * sech(t)
To find the derivative of this function, we will use the product rule.
Derivative of e^t: e^t
Derivative of sech(t): sech(t) * tanh(t)
Applying the product rule, the derivative of f(t) is given by:
f'(t) = (e^t * tanh(t)) + (e^t * sech(t) * tanh(t))
2. f(x) = y^3x * ln(x)
To find the derivative of this function, we will apply the product rule and the chain rule.
Derivative of y^3x: (y^3 * ln(y)) * x^2
Derivative of ln(x): 1/x
Applying the product rule, the derivative of f(x) is given by:
f'(x) = [(y^3 * ln(y)) * x^2] * (1/x) + (y^3 * ln(y)) * 2x
3. f(x) = sinh(x) * tanh(x)
To find the derivative of this function, we will use the product rule.
Derivative of sinh(x): cosh(x)
Derivative of tanh(x): sech^2(x)
Applying the product rule, the derivative of f(x) is given by:
f'(x) = (cosh(x) * tanh(x)) + (sinh(x) * sech^2(x))
4. f(x) = 8sin(5x) * arcsin(x)
To find the derivative of this function, we will use the product rule and the chain rule.
Derivative of sin(5x): 5cos(5x)
Derivative of arcsin(x): 1 / sqrt(1 - x^2)
Applying the product rule, the derivative of f(x) is given by:
f'(x) = (8cos(5x) * arcsin(x)) + (8sin(5x) * (1 / sqrt(1 - x^2)))
These are the derivatives of the given functions.