Derive the function -15^x+x^(-2)+4^x with respect to x

\frac{d}{dx}\left(4^x-5^x+x^{-2}\right)

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Answer

$\frac{-2}{x^{3}}-\sqrt[4]{45}5^x+\frac{\sqrt{17}}{3}4^x$

Step by step solution

Problem

$\frac{d}{dx}\left(4^x-5^x+x^{-2}\right)$
1

The derivative of a sum of two functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(x^{-2}\right)+\frac{d}{dx}\left(-5^x\right)+\frac{d}{dx}\left(4^x\right)$
2

The derivative of a function multiplied by a constant is equal to the constant times the derivative of the function

$\frac{d}{dx}\left(x^{-2}\right)-\frac{d}{dx}\left(5^x\right)+\frac{d}{dx}\left(4^x\right)$
3

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$-2x^{-3}-\frac{d}{dx}\left(5^x\right)+\frac{d}{dx}\left(4^x\right)$
4

Applying the derivative of the exponential function

$-2x^{-3}-1\cdot \sqrt[4]{45}5^x\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(4^x\right)$
5

The derivative of the linear function is equal to $1$

$-2x^{-3}-1\cdot 1\cdot \sqrt[4]{45}5^x+\frac{d}{dx}\left(4^x\right)$
6

Applying the derivative of the exponential function

$-2x^{-3}-1\cdot 1\cdot \sqrt[4]{45}5^x+\frac{\sqrt{17}}{3}4^x\frac{d}{dx}\left(x\right)$
7

The derivative of the linear function is equal to $1$

$-2x^{-3}-1\cdot 1\cdot \sqrt[4]{45}5^x+1\cdot \frac{\sqrt{17}}{3}4^x$
8

Multiply $\frac{\sqrt{17}}{3}$ times $1$

$-2x^{-3}-\sqrt[4]{45}5^x+\frac{\sqrt{17}}{3}4^x$
9

Applying the property of exponents, $\displaystyle a^{-n}=\frac{1}{a^n}$, where $n$ is a number

$-2\frac{1}{x^{3}}-\sqrt[4]{45}5^x+\frac{\sqrt{17}}{3}4^x$
10

Apply the formula: $a\frac{1}{x}$$=\frac{a}{x}$, where $a=-2$ and $x=x^{3}$

$\frac{-2}{x^{3}}-\sqrt[4]{45}5^x+\frac{\sqrt{17}}{3}4^x$

Answer

$\frac{-2}{x^{3}}-\sqrt[4]{45}5^x+\frac{\sqrt{17}}{3}4^x$

Problem Analysis

Main topic:

Differential calculus

Time to solve it:

0.25 seconds

Views:

123