Slope of tangent and derivative of a function are closely related concepts in calculus. In the case of a strict decreasing function, the slope of the tangent line to the graph of the function is always negative. This implies that the derivative of the function is also negative.

Definition of a Strict Decreasing Function:
A function f(x) is said to be strictly decreasing on an interval if for any two values a and b in that interval, where a < b,="" the="" corresponding="" function="" values="" satisfy="" f(a)="" /> f(b).

Explanation:

1. Slope of the Tangent Line:
The slope of a tangent line to a curve at a particular point represents the rate at which the function is changing at that point. In the case of a strict decreasing function, the function values decrease as the input values increase. As a result, the tangent line to the graph of the function will have a negative slope.

2. Derivative of a Function:
The derivative of a function f(x) represents the rate of change of the function with respect to x. Mathematically, it is defined as the limit of the difference quotient as the change in x approaches zero:

f'(x) = lim(h -> 0) [(f(x+h) - f(x))/h]

For a strict decreasing function, as x increases, the function values decrease. This means that the numerator (f(x+h) - f(x)) will be negative for positive values of h. Dividing by a positive value of h will result in a negative difference quotient. Taking the limit as h approaches zero, the derivative of the function will be negative.

Conclusion:
In conclusion, for a strict decreasing function, the slope of the tangent line is always negative and the derivative of the function is also negative. This is because the function values decrease as the input values increase, leading to a negative rate of change.