Answer :
Sure, let's tackle the differentiation of the given functions step by step.
### Problem 3.3: Differentiate from first principles if [tex]\( f(x)=\sqrt[4]{x^6} \)[/tex]
First, we need to simplify the function [tex]\( f(x) = \sqrt[4]{x^6} \)[/tex].
Rewriting the function in exponential form:
[tex]\[ f(x) = x^{6/4} \][/tex]
[tex]\[ f(x) = x^{3/2} \][/tex]
Now we differentiate [tex]\( f(x) = x^{3/2} \)[/tex]:
1. Recall the power rule for differentiation, which states that if [tex]\( f(x) = x^n \)[/tex] where [tex]\( n \)[/tex] is any real number, then [tex]\( f'(x) = nx^{n-1} \)[/tex].
2. For our function [tex]\( f(x) = x^{3/2} \)[/tex]:
[tex]\[ f'(x) = \frac{3}{2} x^{\frac{3}{2} - 1} \][/tex]
3. Simplify the exponent:
[tex]\[ f'(x) = \frac{3}{2} x^{\frac{1}{2}} \][/tex]
4. Rewrite the simplified answer:
[tex]\[ f'(x) = 1.5 x^{0.5} \][/tex]
Thus, the derivative of [tex]\( f(x) = x^{3/2} \)[/tex] is:
[tex]\[ f'(x) = 1.5 x^{0.5} \][/tex]
### Problem 3.4: Differentiate from first principles if [tex]\( f(x) = \frac{2}{x^4} \)[/tex]
First, we need to rewrite the function [tex]\( f(x) \)[/tex] in a form that makes differentiation easier.
Rewriting [tex]\( f(x) = \frac{2}{x^4} \)[/tex] as [tex]\( f(x) = 2x^{-4} \)[/tex]:
Now we differentiate [tex]\( f(x) = 2x^{-4} \)[/tex]:
1. Recall the power rule for differentiation: [tex]\( f(x) = x^n \)[/tex] implies [tex]\( f'(x) = nx^{n-1} \)[/tex].
2. For our function [tex]\( f(x) = 2x^{-4} \)[/tex]:
[tex]\[ f'(x) = 2 \cdot (-4) x^{-4 - 1} \][/tex]
3. Simplify the expression:
[tex]\[ f'(x) = -8 x^{-5} \][/tex]
4. Rewrite the simplified answer in a fraction form:
[tex]\[ f'(x) = -\frac{8}{x^5} \][/tex]
Thus, the derivative of [tex]\( f(x) = 2x^{-4} \)[/tex] is:
[tex]\[ f'(x) = -\frac{8}{x^5} \][/tex]
### Problem 3.3: Differentiate from first principles if [tex]\( f(x)=\sqrt[4]{x^6} \)[/tex]
First, we need to simplify the function [tex]\( f(x) = \sqrt[4]{x^6} \)[/tex].
Rewriting the function in exponential form:
[tex]\[ f(x) = x^{6/4} \][/tex]
[tex]\[ f(x) = x^{3/2} \][/tex]
Now we differentiate [tex]\( f(x) = x^{3/2} \)[/tex]:
1. Recall the power rule for differentiation, which states that if [tex]\( f(x) = x^n \)[/tex] where [tex]\( n \)[/tex] is any real number, then [tex]\( f'(x) = nx^{n-1} \)[/tex].
2. For our function [tex]\( f(x) = x^{3/2} \)[/tex]:
[tex]\[ f'(x) = \frac{3}{2} x^{\frac{3}{2} - 1} \][/tex]
3. Simplify the exponent:
[tex]\[ f'(x) = \frac{3}{2} x^{\frac{1}{2}} \][/tex]
4. Rewrite the simplified answer:
[tex]\[ f'(x) = 1.5 x^{0.5} \][/tex]
Thus, the derivative of [tex]\( f(x) = x^{3/2} \)[/tex] is:
[tex]\[ f'(x) = 1.5 x^{0.5} \][/tex]
### Problem 3.4: Differentiate from first principles if [tex]\( f(x) = \frac{2}{x^4} \)[/tex]
First, we need to rewrite the function [tex]\( f(x) \)[/tex] in a form that makes differentiation easier.
Rewriting [tex]\( f(x) = \frac{2}{x^4} \)[/tex] as [tex]\( f(x) = 2x^{-4} \)[/tex]:
Now we differentiate [tex]\( f(x) = 2x^{-4} \)[/tex]:
1. Recall the power rule for differentiation: [tex]\( f(x) = x^n \)[/tex] implies [tex]\( f'(x) = nx^{n-1} \)[/tex].
2. For our function [tex]\( f(x) = 2x^{-4} \)[/tex]:
[tex]\[ f'(x) = 2 \cdot (-4) x^{-4 - 1} \][/tex]
3. Simplify the expression:
[tex]\[ f'(x) = -8 x^{-5} \][/tex]
4. Rewrite the simplified answer in a fraction form:
[tex]\[ f'(x) = -\frac{8}{x^5} \][/tex]
Thus, the derivative of [tex]\( f(x) = 2x^{-4} \)[/tex] is:
[tex]\[ f'(x) = -\frac{8}{x^5} \][/tex]
