f(t)=5^x sechx
f'(x)=?
Do you mean
f(x) = (5^x)*(sech x), or
f(x) = 5^(x sech x) ?
In the first case, use the differentiation rule for the product of two functions. The second can be solved using the rule for the function of a function
(5^x)*(sech x)
To find the derivative of the function f(x) = 5^x sech(x), we can use the product rule and chain rule.
The product rule states that if we have a function of the form h(x) = f(x)g(x), then its derivative is given by h'(x) = f'(x)g(x) + f(x)g'(x).
In this case, let's consider f(x) = 5^x and g(x) = sech(x).
The derivative of f(x) = 5^x can be found using the chain rule. The chain rule states that if we have a composite function h(x) = f(g(x)), then its derivative is given by h'(x) = f'(g(x)) * g'(x).
For f(x) = 5^x, the derivative is f'(x) = ln(5) * 5^x, since the derivative of 5^x with respect to x is ln(5) * 5^x.
To find the derivative of g(x) = sech(x), we again use the chain rule. The derivative of sech(x) can be expressed in terms of hyperbolic functions as sech(x) = 1/cosh(x), where cosh(x) is the hyperbolic cosine function. Taking the derivative, we have g'(x) = -(sech(x) * tanh(x)).
Now, we can apply the product rule to find the derivative of f(x) = 5^x sech(x):
f'(x) = f'(x)g(x) + f(x)g'(x)
= (ln(5) * 5^x) * sech(x) + 5^x * (-(sech(x) * tanh(x)))
= ln(5) * 5^x * sech(x) - 5^x * sech(x) * tanh(x)
Thus, the derivative of f(x) = 5^x sech(x) is given by f'(x) = ln(5) * 5^x * sech(x) - 5^x * sech(x) * tanh(x).