Derivatives are a fundamental we use in calculus and in our daily lives, some examples are trying to predict stock markets, currencies such as bitcoin, and interest rates. The derivative of a function at a single point, chosen at an input value if it exists, is the slope of the tangent line. To find the slope of a tangent line we must use the formula
An example for the usage of this formula would be finding the slope of the tangent line to the graph f(x) = x^2 at x=4. We would plug in our numbers so it would look like Mtan = lim x-->4 ((f(x)-f(4))/x-4), then it would be Mtan = lim x-->4 ((x^2-16)/x-4), then we would factor it, Mtan = lim x-->4 (((x-4)(x+4))/(x-4)) we would cancel the x-4 and would be left with x+4 so we then can plug in 4 and we would get 8. Therefore the slope of the tangent line at the graph f(x)= x^2 at x=4 is equal to 8.
Another example of the usage of derivatives is finding the derivative of a sine function. This can help trying to predict the stock market as it can drop or rise at anytime and the usage of sine and cosine functions is very crucial. For example when finding the derivative of a sine function we first must understand that the derivatives of sin is cosx and cosine is -sinx
For example, lets find the derivative of f(x) = 5x^3 sinx, with this function we can use the power rule, we get f'(x) (d/dx)(5x^3) * sinx + (d/dx) (sinx) * 5x^3, once we derive it our final answer is f'(x) = 15x^2*sinx+5x^3*cosx.
Lastly the derivative of an inverse function, to put it into simple terms when we have f^-1(x) and want to find the derivative of it we can do dy/dx = d/dx(f^-1(x)) = (f^-1) = 1/f'(f^-1(x)). And if we want to make it even more simpler we can defind g(x) to be the inverse of f(x) so the formula would look as simple as this g(x) = 1/f'(g(x)).