What does it mean if the derivative of a function at a point is exactly 0?

The function has a horizontal tangent line at that point.

The function's slope increases without bound.

The function's slope becomes negative.

The function has no extremum point.

What is the derivative of $f(x) = \sin(x)$ at $x = \frac{\pi}{2}$?

$\cos\left(\frac{\pi}{2}\right)$

Find the derivative with respect to $x$ for $f(x) = (\ln{x})^5$.

$f'(x) = \frac{5(\ln{x})^4}{x}$

$f'(x) = \frac{(\ln{x})^6}{6x}$

$f'(X) = \frac{(\ln{x})^5 - \lim{(\ln{x})^4}}{}}

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2.7 Derivatives of cos x, sinx, e^x, and ln x

4 min read•february 15, 2024

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Now that you’ve learned how to find derivatives of polynomial equations, it’s time to learn how to find derivatives of special functions including $\sin x$ , $\cos x$ , $e^x$ , and $\ln x$ . Finding these derivatives are relatively simple as long as you can remember the rules. 👍

😎 Derivatives of Special Functions

Before we get into each individual rule, here’s a quick table summarizing them.

Function	Derivative
Sine Function: $f(x) =\sin x$	$f'(x)=\cos x$
Cosine Function: $g(x)=\cos x$	$g'(x)=-\sin x$
Exponential Function: $h(x)= e^x$	$h'(x)= e^x$
Natural Logarithm Function: $k(x)=\ln x$	$k'(x) = \frac{1}{x}$

Derivative of $\sin x$

The derivative of $\sin x$ will always be $\cos x$ . Let’s look at an example:

f(x) = 4\sin x +3x

When finding the derivative of this equation, we need to find the derivative of $4\sin x$ and $3x$ separately.

Since the derivative of $\sin x = \cos x$ , the derivative of the first part of the equation is $4\cos x$ . The derivative of $3x$ is $3$ . Therefore $f'(x) = 4cosx+3$ .

Derivative of $\cos x$

The derivative of $\cos x$ will always be $-\sin x$ . Let’s look at an example:

f(x) = 2\cos x +3

When finding the derivative of this equation, we need to find the derivative of $2\cos x$ and $3$ separately.

To find the derivative of $2\cos x$ , we need to know that the derivative of $\cos x$ is $-\sin x$ . Therefore, the derivative of the first part of the equation is $-2\sin x$ . The derivative of 3 is 0, as explained in an earlier lesson discussing the constant rule. Therefore the derivative of the above equation is $-2\sin x$ .

Derivative of $e^x$

This one is pretty straightforward. The derivative of $e^x$ is simply… $e^x$ ! That’s right, the derivative of $e^x$ is just itself. 🤯

Here’s an example:

f(x) = e^x+3x^4

The derivative of the first part of the equation is $e^x$ , since we just stated that the derivative of $e^x$ is itself. The derivate of the second part of the equation is $12x^3$ , according to the power rule. Therefore, $f'(x) = e^x + 12x^3$ .

Derivative of $\ln x$

The derivative of $\ln x$ is $\frac {1}{x}$ . Let’s look at an example:

f(x) = 5\ln x + 2x

The derivative of the first part of the equation is $\frac {5}{x}$ since we know that the derivative of $\ln x$ is $\frac {1}{x}$ . The derivative of $2x$ = $2$ , so $f'(x)=\frac {5}{x}+2$ .

These rules take a little bit of practice, but once you memorize them, it gets simpler! You got this. 🍀

Key Terms to Review (8)

1/x

: 1/x is a rational function that represents an inverse relationship between two variables, where y decreases as x increases and vice versa.

2e^(2x)

: 2e^(2x) is an exponential function that represents the growth or decay of a quantity at a rate of 2 times the base of natural logarithm, e, raised to the power of 2x.

Derivatives

: Derivatives are the rates at which quantities change. They measure how a function behaves as its input (x-value) changes.

e^x

: The exponential function e^x represents continuous growth or decay over time. It is defined as raising Euler's number (approximately 2.71828) to the power of x.

Exponential Function

: A mathematical function where the independent variable appears in an exponent, resulting in rapid growth or decay.

f'(x)

: The derivative of a function f(x) represents the rate at which the function is changing at any given point. It measures the slope of the tangent line to the graph of the function.

Natural logarithm function

: The natural logarithm function, denoted as ln x, is the inverse of the exponential function with base e. It gives the value of y such that e raised to the power of y equals x.

sin x

: Sin x refers to the trigonometric sine function, which relates an angle in a right triangle to the ratio of the length of the side opposite that angle to the length of the hypotenuse.