Question

\(f\left(x^{2}\right)=x^{4}+x^{3}+1\) Put \(x=x^{2}\) \(f\left(x^{4}\right)=x^{8}+x^{6}+1\) \(f^{\prime}\left(x^{4}\right)=\frac{8 x^{7}+6 x^{5}}{4 x^{3}}\) \(=2 x^{4}+\frac{3}{2} x^{2}\)

Short answer

Answer: The expression for \(f\left(x^4\right)\) is \(x^{16} + x^{12} + 1\), and its derivative \(f^{\prime}\left(x^4\right)\) is \(x^{11}(16x^4 + 12)\).

Step-by-step solution

Perform the substitution

Since we are given the substitution \(x = x^2\), we can rewrite the original function as \(f(x) = (x^2)^4 + (x^2)^3 + 1\). Now, we can substitute \(y = x^2\) into the expression as \(f(y) = y^4 + y^3 + 1\). Finally, we substitute \(y = x^4\) to obtain the expression for \(f\left(x^4\right)\).

Apply the substitution

Substitute \(y = x^4\) into the expression for \(f(y)\): $$ f\left(x^4\right) = (x^4)^4 + (x^4)^3 + 1 $$

Simplify the expression

Calculate the powers to obtain: $$ f\left(x^4\right) = x^{16} + x^{12} + 1 $$

Compute the derivative of \(f\left(x^4\right)\)

Find \(f^{\prime}\left(x^4\right)\) by differentiating the terms with respect to \(x\): $$ f^{\prime}\left(x^4\right) = \frac{d}{dx}(x^{16} + x^{12} + 1) $$

Apply the Chain Rule

Differentiate with respect to \(x\), applying the chain rule: $$ f^{\prime}\left(x^4\right) = 16x^{15} + 12x^{11} $$

Simplify the derivative

Factor out the common term \(x^{11}\) to simplify the expression: $$ f^{\prime}\left(x^4\right) = x^{11}(16x^4 + 12) $$ Now we have found the expression for \(f\left(x^4\right)\) and its derivative \(f^{\prime}\left(x^4\right)\).