How to find the derivative for these equations?

Question

Ezra Adams · Accepted Answer

#-12e^(-4x)+e^x-2x#

Re-writing:

#3(e^x)^-4+e^x-x^2#

We know #dy/dxe^x=e^x#

Using the Chain Rule:

#dy/dx=dy/(du)*(du)/dx#

Letting #u=e^x#

#dy/dx(3(e^x)^-4+e^x-x^2)#

#= (-4)3(e^x)^-5*e^x+e^x-2x# *

#=-12e^(-4x)+e^x-2x#

* #(e^x)^-5*e^x=e^(-5x+x)=e^(-4x)#

Aurora Alvarado · Answer

undefined To find the derivative of a function $f(x)$ with respect to $x$, you can use differentiation rules. The specific method depends on the type of function given. Here are the general steps for finding derivatives:

1. **Polynomial Functions**: For a polynomial function $f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0$, use the power rule: $f'(x) = n \cdot a_n x^{n-1}$.

2. **Exponential Functions**: For an exponential function $f(x) = a \cdot e^{bx}$, the derivative is $f'(x) = a \cdot b \cdot e^{bx}$.

3. **Trigonometric Functions**: For trigonometric functions like sine, cosine, and tangent, you can use the trigonometric derivative rules. For example, if $f(x) = \sin(x)$, then $f'(x) = \cos(x)$.

4. **Product Rule**: If $f(x) = u(x) \cdot v(x)$, then $f'(x) = u'(x) \cdot v(x) + u(x) \cdot v'(x)$.

5. **Quotient Rule**: If $f(x) = \frac{u(x)}{v(x)}$, then $f'(x) = \frac{u'(x) \cdot v(x) - u(x) \cdot v'(x)}{[v(x)]^2}$.

6. **Chain Rule**: If $f(x) = g(h(x))$, then $f'(x) = g'(h(x)) \cdot h'(x)$.

Apply these rules according to the type of function you're dealing with to find the derivative.