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