can cancel the h in the numerator and the denominator to get f′(8) = . Now,we take the limit: f′(8) = = .12.
We find the derivative of a function, f(x), using the definition of the derivative, which is f′(x) = . Here f(x) = sin x and x = . This means that and
. If we now plug these into the definition of the derivative, we get
= = . Notice that if we now take the limit, we getthe indeterminate form . We cannot eliminate this problem merely by simplifying theexpression the way that we did with a polynomial. Recall that the trigonometric formula sin (A+ B) = sin A cos A + cos A sin B. Here we can rewrite the top expression as = = . We can break up the limit into + = + . Next, factor out of the top of theleft-hand expression: + . Now, we can break this into separate limits: + . The left-hand limit is = = . The right-hand limit is . Therefore, the limit is .
13. 2 x + 1
We find the derivative of a function, f(x), using the definition of the derivative, which is f′(x) = . Here f(x) = x^2 + x and f(x + h) = (x + h)^2 + (x + h) = x^2 + 2xh + h^2 + x +
h. If we now plug these into the definition of the derivative, we get f′(x) = = . This simplifies to f′(x) =