Derivative of exponential, logarithmic and a pow x function

Exponential function

$$f(x)=e^x \Rightarrow f'(x)=e^x$$

The derivative of the exponential function is the exponential itself.

Logarithmic function

$$f(x)=\ln x \Rightarrow f'(x)=\frac{1}{x}$$

$$f(x)=\log_{b} x \Rightarrow f'(x)=\frac{1}{x\cdot \ln b}$$

Functions of the type $a^x, \ a>0$

$$f(x)=a^x \ (a>0) \Rightarrow f'(x)=a^x \ln a$$

In this case we need $a$ to be a positive constant, otherwise the function $f (x)$ would not be derivable.

Let's see examples that include these types of functions and others.

The function:$$f(x)=\sin x + e^x -x^3$$

The derivative is:$$f'(x)=\cos x +e^x - 3x^2$$

The function:$$f(x)=3^x-\cos x+ \ln x$$

The derivative is:$$f'(x)=3^x\ln 3-(-\sin x)+\frac{1}{x}=3^x\ln 3+\sin x +\frac{1}{x}$$

The function: $$f(x)=\log_{10}x +5x^3+3$$

The derivative is:$$f'(x)=\frac{1}{x\cdot \ln 10}+15x^2$$